Just a thought for if its sunny (when it will be busiest)
Drive to the chain-link ferry (Swanage side) and park up
Go over on the ferry on foot and then catch an open-top bus to Poole bus station hoping off at the civil center roundabout for the Parkrun
To return either jump back on a bus or walk across the park to Poole Quay and catch the Brownsea ferry making sure you get the one that stops at Brownsean AND the chainlink ferry. Stop for a stroll on Brownsea if you fancy
Admittedly (probably) a more expensive way to do it, but it’s a nice area and the bus and boat trip options will show you some nice bits 🙂