Clothes horse - much better name than a clothes drying rack - I like.
I imagine you are talking about a baseboard heater or radiator, which actually in operation heat a room by convection and very little by means of radiation.
What happens is as follows:
- the air in your room has a certain humidity level, which means the amount of moisture in the air
- 100% relative humidity means that the air is completely saturated with moisture and can hold no more. If your room is 40% relative humidity, clothing will still dry.
- and as the air warmed by the radiator is able to hold more moisture - the relative humidity of the warmed air has fallen to a lower level.
- the warmer air, now a little "dryer" can accept moisture from your clothes a little more quickly
- the warmer air rises, accepts moisture, becomes a little more humid, and is continiously replaced by more "dryer" warm air that passes around the clothing.
So the question of whether the orientation of parrallel or perpindicular placement of the clothes horse to the radiator will dry clothes the fastest, is subject to the air circulation pattern set up around the clothes. If one is drying one or two pieces of clothing, then parrallel would be the best as both pieces of clothing are above the radiator. As the number of pieces of clothing increases, more garments ( or parts of the garment ) will be farther from the radiator and not dry as quickly, so I suspect placement, parrallel or perpendicular, would not matter, all things being equal,
On the other hand, the small amount of radiation transmiitted from the radiator to the clothing, would be blocked by parallel placement of the first piece of clothing.
You could test orientation and clothing drying times by timing "test runs" of each orientation and note the length of time it took to when you feel the clothing is dry.