You mention a number of closely related, but different, concepts in your question. The answer is still yes, because every concept you listed does exist in nature; Evolution is pretty good at coming up with any example you can think of. However, they are distinct and separate concepts, so let's address each of them individually.
You didn't mention this yourself, but it was the first comment, and is relevant to the rest of the topics so we'll touch on it first.
Parthenogenesis is "is a natural form of asexual reproduction in which growth and development of embryos occur without fertilization." This means that the creature can have young without having sex, but this does not mean it is born pregnant. In most cases, creatures capable of parthenogenesis will still develop through their normal life cycle and only start the development of their young once they reach maturity. In other words, they may have an egg but they don't yet have an embryo in them at birth. Some species, such as sharks, usually reproduce sexually and only resort to parthenogenesis in the absence of males. Others need triggers to start their pregnancy, such as the commonly quoted example of the whiptail lizard which produces more young if mounted by another lizard to emulate sexual reproduction. Despite technically being defined as asexual, some pathogenic females actually need to mate with a male, using the sperm to trigger the development of young. This is called Gynogenesis
Usually these species are not pregnant at birth, even if they are able to conceive without mating.
I'm using the definition Straight Dope was referring to here, I go more into other definition below.
Paedogenesis is not an example of being pregnant from birth. Instead paedogenesis is "reproduction by sexually mature larvae, usually without fertilization". The key part of this definition is that it's a sexually mature larvae. Larvae are usually considered the early, usually not sexually mature, stage in a life cycle. Usually a larvae needs to go through metamorphosis into an 'adult' stage before they are able to reproduce. Thus a larvae being able to reproduce on its own is noteworthy enough to have a special term to describe it.
Usually paedogenesic species start as having a larval form and adult form, as most insects have. At some point a genetic mutation makes it possible for the larval form to reproduce on its own, usually via parthenogenesis. Paedogenesic species often will eventually stop developing into their adult phase, though they can be triggered to develop into them sometimes. Less commonly, paedogenesis can also involve sexual reproduction.
As with the distinction I made with parthenogenesis, these species are not necessarily born pregnant. Usually they don't develop embryos until a certain stage in life. That stage is usually early due to their never developing out of larval phase, but still embryos usually don't develop pre-birth. As pointed out, a small number will still reproduce sexually and may require mating before the embryo will develop.
Your Wikipedia link for paedogenesis redirects to neoteny, which is a different concept. Neoteny is "the delaying or slowing of the physiological (or somatic) development of an organism, typically an animal". This can be any sort of retention of immature characteristics. Domestic dogs are an example of neoteny because they have been breed to retain many puppy traits into adulthood, mostly because humans find those traits adorable and prefer dogs with those traits.
Since most examples of paedogenesis involve the creature never metamorphosing out of their larval phase, or delaying the metamorphosis much longer, this is an example of neoteny, but most cases of neoteny are not paedogenesis.
The reason Wikipedia redirects to neoteny is because they are working with a slightly different definition of paedogenesis. To quote Wikipedia's neoteny definition:
In progenesis (also called paedogenesis), sexual development is accelerated.
This is an alternate, but less common, interpretation of paedogenesis. Unfortunately many of these terms in biology have a few slightly different meanings, as you will see again shortly. Wikipedia is using a different one to me.
Telescoping Generations and Aphids
Now we're getting somewhere. As you saw, Wikipedia defines telescoping generations as:
characterized by a viviparous female having a daughter growing inside her that is also parthenogenetically pregnant with a daughter cell.
This definition explicitly says that the daughter is already pregnant. So, yes, aphids are born pregnant!
These aphids are clonal. That means that their daughters are identical copies of them, and reproduce via parthenogenesis. They don't have to have sex. Considering this, there is no little need for them to wait. In fact, while a daughter is still growing in her mother, her own eggs will hatch inside of her, so the original mother has her granddaughter developing inside of her.
The advantage of this technique is that it speeds up birthing the next generation, since some of the development time of the third generation overlaps with the development of the second generation. The main downside is that it means the aphids need to quickly find and eat food after their birth, as their own daughters are already well developed and are dedicating additional resources to developing granddaughters. The newly born aphid is not just eating for one, but for many dozens. The other downside is that they are clonal, and all clonal species lack genetic variation which makes them vulnerable to disease and less capable of adapting to changes in environments.
Incest and Cannibalism
Despite the definition of telescopic generations, Wikipedia (and most other dictionaries) use, there is usually another species that is considered as a telescopic generation: Adactylidium.
This is a type of mite that manages telescopic generations without using parthenogenesis. This is a reason I stress parthenogenesis is different from telescopic generations. With this species, the mother will be pregnant with one son and multiple daughters. The son will mate with his sisters while still in the mother, impregnating them before they are even born. The daughters will eventually eat their way out of the mother, killing her in the process, and leave their brother/baby-daddy to die in the carcass of their mother. The newly born daughters are already pregnant and will spend 4 days looking for food (a specific type of egg) while waiting for their own incestuous daughters/nieces (who are in turn already pregnant with the hosts grandchildren) to one day kill them via their cannibalism/birth.
Believe it or no,t this can be seen as advantageous for diversity compared to the aphids. It still retains some form of sexual reproduction. By mating with her brother, the mother has ensured each daughter will get different sperm and thus will have slightly different genes. The difference isn't large, since there isn't much genetic variety left after numerous generations of obligatory incest, but any degree of genetic variety will allow some degree of protection against disease or changing environmental pressures. The mere presence of sexual reproduction increases the odds of mutation, which does increase the odds of being able to adapt to different situations. So the Adactylidium gets the evolutionary thumbs up of approval for incest and cannibalism.
Other evolutionary fun
This is just a start of all the interesting types of evolution we have. A quick list of other interesting things that come up.
Sharks also commit cannibalism in the womb
Bed bugs have an even ickier means of pregnancy, with their aptly-named traumatic insemination, a technique that numerous separate species all evolved independently.
Fruit Flies are one of the very tiny number of species that preferentially prefer incest with their brothers over other options, as opposed to begrudgingly allowing it because its too hard to avoid. The advantage of the female's children having 3/4 of her genetics instead of 1/2 is just enough to compensate for the harm incest causes to the success of the kids.
Male ants have no father, only a grandfather.
Some frogs will keep half as much male DNA, resulting in all kinds off odd hybrid situations
I could go on, but were leave it there for now. Evolution can lead to some interesting things.