I think parking:$side:zone probably does have a place even where other parking tags haven't (yet) been set. 

Firstly, there's a traffic order which includes that street in a CPZ, which is the only tag we currently have for it. Maybe a relation would be better, but none has yet been proposed for this purpose.

Secondly, when other parking tags are added from StreetComplete survey or aerial imagery, it will make it a lot easier to set and keep up to date the parking conditional access and restriction tags for marked bays (and possibly single yellow lines). I haven't actually started doing this yet, but will do soon.

Finally, even where waiting is prohibited at all times, many parking authorities print the CPZ after the street name on the penalty charge notice.

Waltham Forest recently reissued their parking-related restrictions in a single order, which has allowed me to tag almost all the CPZ streets there. In Newham, it involves a lot more digging around in The Gazette.