Can a person be on probation and parole at the same time?

Yes, however, it's very rare.

Under general circumstances, if the individual is on parole, then they've just been released from a state correctional facility. For the individual to be put on probation (after release), it means that the parolee committed a new crime, which means the parolee also violated the parole agreement.

If this were to happen, usually, the parole would be revoked and the individual would go back to prison.

However, in rare circumstances, the parole board will give the parolee "another chance", and not send the parolee to prison.