I encountered (a while back) that occasionally parking conditions may be different on the carriageway (i.e. the street) and on the kerb. Take for example this sign combination combining on-kerb disc parking with “no stopping” on the road itself:
I don’t think you gotta need to tag the no_stopping for the lane, since having street-side parking literally means: no parking on the lane, only street-side.
In this case I agree with @Nadjita – simply tag the on-kerb parking since it implies that there is no parking on the lane. In more complicated cases (e.g. street-side parking bays with larger distances in between, where parking can take place on the carriageway) or if you need it more precise, for “multiple rules” there is only the possibility to map the “carriageway-related” rule on the road center line and the “side-space-related” rule to separately mapped parking areas.