If your question is "is Barron Park a good area for Stanford University" most definitely yes. You can bike from Barron Park to Stanford, and the Margarita bus (stanford's bus) picks up from very close by. Easy!
It is relatively challenging to rent in Palo Alto and I recommend you go to the rental on the morning it is listed, as quickly as possible, in person, and make a case for yourself. Have a pay stub or proof of "how you can pay rent" in hand.