When is the best month to avoid a lot of tourists? June or September specifically?

I would say September.

September when the kids are back to school, plus cooler for outdoor activities

To answer your question Septembers a better month to avoid tourists.

Mid September. June tends to be BUSY

Paul Turner is very WRONG, October most all tourist are gone except the Chinese!

September

Wintrr months and fall is less crowded. Summer and spring, the park is full. Especially spring break.

Early April is the time we went less crowd. In the past we went in May, September

Avoid June