Can a person on federal probation move while on supervised release?

You would need to obtain permission from the Court/probation officer. Generally, an individual who is on supervised release should never change residence without permission/notifying but I can't advise you since I am not familiar with the facts of your case nor do I practice in Pennsylvania.

Therefore I recommend you contact the attorney who originally represented you, or consult a local attorney in Pennsylvania, or the local Federal Public Defender to make sure that you handle the matter properly. This is very important.

First of all, I think that you are confusing terms. Federal probation is a term applied to people who have not served a prison sentence, whereas supervised release (the analogous term to what used to be called parole) is for people who have served a prison sentence. Having said that, whether you are on probation or supervised release, you cannot move unless your supervision is transferred to California. The probation office and AUSA must be consulted, and the judge must approve any such transfer. My suggestion is that you consult with a lawyer who does federal work in Pennsylvania.

(Please note that this response does not constitute legal advice nor does it establish an attorney-client relationship.)

I agree with the assessments of both of my colleagues, regardless of whether you are on supervised release or pobation. Either contact your prior lawyer, or consult with an experienced criminal defense lawyer licensed to practice law in PA to get the permission you need to transfer the supervision to CA before you move. Good luck.

DISCLAIMER
This answer is provided solely for informational purposes, for you to use as a starting point when speaking directly with a lawyer in your State. I do not practice law in your State, and this answer does not constitute legal advice, create an attorney-client relationship, or constitute attorney advertising. I urge you to immediately contact an experienced criminal defense lawyer admitted to practice law in your State before you make any decisions about this case.