How restricting it is depends a lot on what you want to do while there. No problem at all if you just want to drive around and sight see. Your dog will have free run of your own vehicle of course and there are plenty of places he can get out for 'nature calls'. What he will NOT be able to do is walk with you on hiking trails or visit any of the parks buildings (visitor centers etc. restaurants) but campgrounds should be no issue. The park has never been very busy during winter months either of the times I visited then making it a nice time to visit.