The animal mascots that the Partridge family is dancing with are from The Banana Splits Adventure Hour (1968).
At the Kings Island Inn & Conference Center, where Keith is preparing to go swimming---and the adults order their alcohol---the waiter is Cincinnati Reds' catcher Johnny Bench.
This episode was filmed at King Island in Cincinnati, Ohio. Brady Bunch episode The Cincinnati Kids (1973) was filmed at the same amusement park 11 months later.
Shirley Jones (Shirley Partridge) breaks the fourth wall in this episode. After Chris and Tracy find her at the "Lost Parents" shed, she shrugs her shoulders at the camera.