Parole cannot keep you on parole if payments are still outstanding. However, they can require you to make timely payments. They may also not terminate you early due to the outstanding balance.
Seth Weinstein, Esq.
Practicing Criminal Defense in Southern California
This reply should NOT be considered a legal opinion of your case / inquiry. At this time I do not have sufficient factual/legal documentation to give a complete answer to your question and there may be more to the issues you raised then I have set out in my brief reply.
Assuming you have otherwise complied, without a hiccup, with parole ---- If you reached an agreement which will have you making payments for the rest of your life, paying it off can't be a condition of parole. Parole can require that you make regular, timely payments, however.
As to whether you can get an early kick from parole, that is most definitely fact-based and turns on your relationship with BPT and your agent. Discuss it with him/her. You won't know until you ask.