This seems to come down to what you mean by "pierce".
I have no familiarity with the procedure, but I think I've been able to find plenty of evidence online indicating that the video is a somewhat accurate depiction of something that commonly happens a a step in the IUD insertion process. Obviously, the video does not show a real cervix.
The tool that is used appears to be a single-tooth tenaculum, a real surgical instrument that is really used in the placement of intrauterine devices. Describing it as "piercing the cervix" may be a bit imprecise or unclear wording. The hooks on the end are pushed into the flesh of the cervix in order to grip or clamp the cervix. It seems that this often causes bleeding, so I'd guess it may create superficial punctures, but it does not create a puncture that goes through to the uterus.
The purpose of the tenaculum is to allow the physician to control the position of the cervix during the procedure. It appears to be a recommended although not essential step in IUD insertion.
E.g. per "Insertion and Removal of Intrauterine Devices", Brett Andrew Johnson, Methodist Charlton Medical Center, Dallas, Texas, Am Fam Physician. 2005 Jan 1;71(1):95-102.
The physician should stabilize the cervix during the insertion of the IUD with a tenaculum
Per the abstract of "Effect of cervical traction with a tenaculum on the uterocervical angle", NICHOLAS JOHNSON and DAVID R. BROMHAM, March 1991:
Moderate cervical traction straightens the uterus and the routine use of a tenaculum theoretically makes insertion of an intrauterine device safer
The use of the tenaculum is associated with pain and bleeding1; these effects appear to be frequent,2 which seems not very surprising for a device that functions by poking the cervix with sharp hooks.
There are alternative clamp styles (atraumatic tenaculum, Allis clamp, the suction cervical stabilizer that is being developed by Aspivix, the company that made the cited blog post) but I don't have much information on how they compare medically.
The IUD is not inserted into the uterus through the puncture wounds from the tenaculum. The cervical canal is sounded and an insertion device is used to pass the IUD through the cervical canal.3
That reddit thread itself seems to have several comments with essentially the same information about this; e.g.
New Data on Atraumatic Tenaculum Pain Scores, Ob/Gyn Clinical January 1, 2016
"Tenaculum: For Over 100 Years Women Have Endured Pain in Gynecology", blog post, Aspivix