This is something of an urban myth - the drying time gains were made by using oven-dried enamel coatings on the parts that could withstand the required curing temperatures.
The so-called "baked enamel" coating used by Ford was called "Japan Black" (produced for Ford by APV Engineered Coatings) - this is an asphalt-based enamel coating which is applied to the desired object and then baked at around 400F for an hour or so. The color was provided partly by the asphalt base used in this particular enamel and partly by the addition of carbon black pigment. You could technically add other pigments to it in place of the carbon black that Ford used but ultimately it's always going to come out pretty dark.
Of course other baked enamels were available and could be dried in comparable times - but at the time other pigments were far more expensive, and what's more were massively prone to fading (sometimes lasting only a few years). Add these factors together and black just makes the most sense (interestingly there's some anecdotal evidence that suggests Japan Black provided better damp-proofing than other contempory enemals but I have no idea if Ford were aware of this when choosing it).
Trent Boggess (Professor of Economics at Plymouth State University and antique Ford enthusiast who has also spent time working at the Henry Ford Museum) wrote an excellent article detailing much about the paints, materials and processes used during the so called "black years" of the Model T. I won't reprint the entire thing here but the pertinent paragraphs are:
There appear to be several good reasons for the choice of black as the color of the paint. First, black color varnish paints tended to be more durable than lighter colored paints. Authorities on paint in the 1920's noted that black paint tended to last longer than paints with lighter colored pigments. Second, as mentioned above, the addition of Gilsonite improved the damp resisting properties and the final gloss of the paint, but also resulted in a very dark colored paint. The range of colors that asphaltum paints can have is quite limited. The dark color of the Gilsonite limits the color of the final paint to dark shades of maroon, blue, green or black. Cost may also have been a factor. The carbon black pigment used in these paints is probably the least expensive pigment available; almost any other pigment is more expensive than carbon black. One often cited reason for the use of Japan black on the Model T was that it allegedly dried faster than any other paint. However, there is no evidence in either the Ford engineering records or the contemporary literature on paint, to indicate that that was the case. The drying time of oven baking Japan black is no different from the drying time of other colored oven baking paints of the period. In short, Model T's were not painted black because black dried faster. Black was chosen because it was cheap and it was very durable.
The claim that black was chosen because it dried faster than any other color is not supported by the Ford engineering documents, the contemporary literature, nor by the first hand accounts of Ford Motor Company employees.