Local authorities in Spain do have the authority to enact their own law-ish regulations, which are called 'ordenanzas'. For example, if I remember correctly, motorbikes are allowed to park on the pavement by default in Barcelona unless a sign says otherwise, but it is forbidden in Madrid unless a sign explicitly allows it.

I think local government in Spain has at least as much authority as it does in the UK, maybe more, but almost certainly less than it does in the US.