Two examples are near me in the form of Yosemite National Park and Sequoia-Kings Canyon National Park.
Yes, if you follow State Highways into any of these parks you will have to pay an entrance fee. With Yosemite the Old Coulterville Road is accessible to high clearance vehicles and will take you into Yosemite Valley via Foresta. In theory Foresta Road isn't abandoned to El Portal and hypothetically could be repaired in the future to a condition accessible to vehicles. Chowchilla Mountain Road once reached Wawona but appears to be closed now about a mile or from the Yosemite park boundary aside from Fire Evacuation.
With both Sequoia and Kings Canyon the Generals Highway can be reached between both parks via Sequoia National Forest Road 14S75. This will take you up past the Redwood Mountain Trail and land at Generals Highway near the Kings Canyon Vista. I would say 14S75 is a high clearance road but it is rough and had some damage above Eshom Valley.
When the Mt. Blue Sky (née Mt. Evans) Road re-opens, you can start driving before 6:00 AM and there's no fee. Similar in Rocky Mountain National Park (and probably others).
In an episode of Bill Nye's old show he drove a boat car. That could be used on a literal island to avoid bridge or ferry tolls.
I believe there's a secret entrance to Stone Mountain.