Many airports have parallel runways but only one runway has a PAPI/VASI. If both runways start at the same point and have similar touch-down points, is it safe to use the parallel runway's lights for the approach and landing?
Per the AIM (2-1-2 - Visual Glideslope Indicators)
The visual glide path of the PAPI typically provides safe obstruction clearance within plus or minus 10 degrees of the extended runway centerline and to 4 SM from the runway threshold.
(Deviations from that are supposed to be noted in the Airport/Facility Directory, for example if the clearance path is offset from the centerline, or theoretically if you could use a parallel PAPI...)
A few miles out you're probably within the 10 degree arc, so the PAPI will be useful, but as you get closer that arc narrows, and as Lnafziger mentioned there might be an obstacle in your way (like the FAA-Standard 50-foot tree right at the threshold line).
Also as abelenky pointed out you're not supposed to be using parallel PAPIs/VASIs for approach guidance - It's a good cross-check, but not a primary reference.
Not necessarily. The PAPI is installed, and the angle is set for the obstacles in the approach path for that specific runway. There could be obstacles on the parallel that the other PAPI would not give you clearance over because it was never surveyed for that.
It is considered "safe" for a pilot to use all available information to safely conduct a flight. This does not exclude using other references, such as a parallel PAPI as a means to cross-check.
However, it is not legal to use a parallel PAPI as your primary guidance. If there were to be an incident, and you were to tell an investigator, "the parallel PAPI indicated that I was on the GS", he'd think you were a bit nutty, and fault you.
If instead, you were able to tell him "I was properly lined up for the Left, and then cross-checked with the PAPI on the Right, which also indicated a stabilized approach on the GS", he might commend you for creative cross-checking.