Is there a way I can drive there from the bronx without paying the tolls?

There was a time it was free from The Bronx. After the park's renovation, the city placed the off ramp after the tolls. Parks revenue

Unfortunately thats a no. Only way is through is that toll bridge. But, what you can do is park in the city, and just take the bus to Randall's Island, which is currently free. Thats the only way to avoid toll.