Appropriate Indoor Temperatures For Parrotlets, Drafts

I live in San Mateo, CA, and my apartment typically ranges from 60F
to 80F all year round. The upper and lower ends of this range are
slightly outside the parrotlet's recommended environment- is that dangerous?

Not sure where the parameters came from - I know people that keep parrotlets out of doors in places as diverse as Florida where it is very hot and humid to CO where it can snow. Certainly any environment you are comfortable in will be fine for a pet parrotlet, particularly if it is indoors. They can withstand a wide variety of temperature ranges if they are healthy and acclimated to them.

Is it safe to put my parrotlet's cage near an open (but screened)
window, or are potential drafts too dangerous?

I have no idea what kind of 'draft' your bird would be subjected to as I don't live where you do. I do know that air conditioning vents can have detrimental effects if they blow directly on the bird and if you are close to the ocean with damp, cold fog, that could be a problem but this are birds designed by nature to live out of doors. A breeze isn't going to be a problem for an otherwise healthy parrotlet.