I see that this bug report hasn't received activity since 2006. I am still experiencing this issue in 2024 with zsh 5.9, where jobs -p is outputting more than just the PIDs. I think the most optimal solution is for zsh to update the output of jobs -p to be POSIX-compliant and match bash. However, if the zsh developers do not want to break compatibility, another option is to add a new flag that has the correct behavior. For example, a capitalized version, -P.
Thanks,Aaron Franke