Just getting back to the general issue, and taking Sir H’s point of not causing problems where none really exist, most by-laws related to vistor parking come under a provision that you can’t do it without written permission.
So if you’re in a situation where there are plenty of spaces and residents want to use them, how about asking them to apply for permission and even charge a nominal rent (slightly less than the going rate for car spaces in the area) for doing so.
Then in the future, when the lack of visitor spaces becomes an issue, you can start pulling back on the permissions on a last-in, first out basis. These spaces are common property and they are an asset – use them wisely.