When parity is fixed at the beginning, you limit your moves to do the F2L.
I don't understand. Please explain.
I think what Ton means is that if you would do parity last you can do cross without adjusting D-layer afterwards. If you had an easy pair that you could insert with RUR' you could just do that directly. If you would know you had parity you might do a D or D' after building the cross to solve parity which means you couldn't just insert that pair with RUR' (you could still insert it though)
I don't think Ton meant to say that you would be limiting any of the 18 possible face-turns or even restricting to an 3OP-blind-like restrictive set of moves.
Or maybe Ton was mistakenly thinking more about the
Barrel/Octagon that has a similar parity, but can be easier to solve by taking advantage of the fact that edges and corners can be in multiple solved positions.