In short, the answer is you only get the pass travel benefits for the carrier you are affiliated with. If you are awarded a seat flying for Pinnacle, you will get the Delta pass travel benefits only.

Just to be clear on Delta pass travel: non-wholly owned regionals flying under the Delta Connection brand do not receive buddy passes (with the exception of Skywest). Parents travel at a yield fare, but are pretty much stuck to the bottom of the standby priority list.

You and your family will be able to also interline on other carriers using ZED fares if you need to. Sometimes these are restricted to being employed for at least 3, 6, or 12 months.