What could be causing low water pressure in cold line in only part of the house?
Somewhere between the good pressure and bad pressure there's a partial clog in the cold water line. Finding it will be tedious. You don't mention pipe material, which makes speculating about what it is difficult - if it's only happened in the past year and no work was done about the time it started, you'll just have to hunt. If plumbing work was done about that time, start there. If you happen to have galvanized iron pipe, think hard about total replacement, IMHO.

Possible clogs include - excessive pipe seal tape on threaded joints getting loose in the pipes - lumps of corrosion breaking loose and jamming downstream - could be corrosion from a waterline outside your house - Sand from a well running low... dissolved materials coming out of solution in the pipes, suspended materials settling in the pipes. 

My personal most annoying was a soldered copper elbow that was almost, but not quite, full of solder on the inside (and evidently had been for years.) Great pressure for half a second, followed by a trickle. I just disassembled the pipe until I found it, when I finally faced the agony of fixing it.