Psy.D. and Ph.D. programs both tend to have application deadlines in December and will begin in the fall semester (usually August or early September), yes. It makes sense for a few reasons: 1) They typically receive a relatively large number of applications for a very small number of spots, so that 2) screening these applications takes a good bit of time, and 3) inviting individuals for interviews knocks out about a month's worth of time by itself.
Typically, the incoming class roster will have been determined by late April, which leaves students with the summer to prepare for their move (e.g., pack, find a place to live, tie up loose ends at home, etc.).
If there are any programs that begin in any semester other than the fall, then they are very, very few and far between. As in, I don't know of any off the top of my head.