It's nothing to do with free movement, it's about the EU deciding to charge tourists. The US does not have free movement with the UK yet Americans can come here for tourism purposes Visa free for up to 6 months.
If Brussels wants to charge UK tourists then that's up to them. The real question is how this will work re: the Common Travel Area between the UK and Ireland. If Brits travelling from NI to Dublin have to pay £6 then Sinn Fein will throw a tantrum and EU politicians can probably expect a visit from the IRA. Make sure you check under your cars lads..
So how much time is left until a deal must have been signed in time for the parliaments to approve it?
Do you mean the withdrawal agreement or the trade deal?