As far as can tell I can obtain a passport, the real question is getting permission to travel outside the country. So my question is:

Can a California Parole Officer grant a California Parolee permission to leave the country for a short amount of time and return?

As long as the rules of your parole, allows you to have a passport. (some rules require you to surrender, so ask to verify your rules.

Also it depends on your level of parole, and if there were any rules put on you, but for a very good reason (vacation is not one) they could, funeral of close family member or something along that line

Like they say," look in the horses mouth". Why not just ask your parole officer?