Is it alright to shave my private parts in the gym's showers?

No. People will look at you in the wrong way and you may start people to avoid you everywhere inside the gym. They just want to take a shower, not see a guy shave himself. Would you mind if you saw a guy shave his but? If you answer no, then that is the answer you should consider people may feel about seeing you shaving yourself inside the shower area. Hope this helps.