Ldij gives the "definitive" answer, but if you want to know what the parole officer is permitted to approve, with certain exceptions the place of residence has to be in the county where they were residing when they committed the offense. The exceptions are that they may be required to live elsewhere if there are issues with the victim's safety or that there's a better chance (treatment program, family support, a job) of the succeeding elsewhere. Such moves have to be specifically arranged between the departments in the two counties involved, so it can take some time.
As to approval of this really depends on what supervision level the offender is on. In some cases, all it takes is a phone call to the PO. In others, it gets more involved.
In addition to a home, they can indeed be placed in a halfway house or a "transitional treatment center" (if such is requierd). To get another "residence" approved, the process used to approve the first one must be followed: whether there is any adverse history of the parolee with regard to that place, who lives there and what their relationship is with the parolee, if they have criminal history, even if there are certain animals present. They'll also check to see if there is anything else in the neighborhood that could be a concern. Again, this background investigation can take time.