If someone finds a build list for a gaming PC on a site like pcpartpicker, should they buy all the parts roughly the same time? Or could they space buying the parts out over several months, without having to worry about them becoming unavailable or outdated?

Depends on what parts. Some may become outdated, some may become high-demand and not easy to get. For the most part (heh), you'll probably be fine though.

Ahem.....

ok let's start with the GPU first, this one is already outdated before you even bought it.

CPU is outdated, once you are at home, installed everything and finally start to enjoy the new computer.

The only thins which do not have such a high rate is like the motherboard/RAM and HDD......

The main question is: What are YOU looking for?

Yes, you might have to take a different GPU, if you have chosen a low end model, but If you have chosen a mid-range model, it might become a low price model, buy the time you buy it. Nothing to worry about, so long your games aren't changing too much ;)

And for a more precise answer, more info needed ;)