Why is my parking break lever not staying in the off position.

There is an adjustment on the cable. Since you replaced everything you may need to relax the cable slightly to make sure it stays in the rest position. Alex

Replacing those parts should have put a little more tension on the park brake cables and in turn the park brake lever - that should make the lever 'less likely' to “pop up” as it would tend to pull the lever downward .. It is difficult to determine here online.. Vehicle will need to be hoisted for proper inspection of the cabling *system* and related park brake components.