What if we kept them, but made them public?
Sacramento does this, they manage the lots owned by the local SEIU chapter and charge for them at times when the SEIU does not need the parking. (https://www.cityofsacramento.org/Public-Works/Parking-Services/Lots-and-Garages/Rates -- this applies to the SEIU lot but there may be others). I do wish more cities or jurisdictions attempted to do this but I imagine there's liability/insurance issues that are much easier to overcome when you can charge for that parking like they can here. May be harder to do as free parking.
We know a non-profit theater company that is attempting to work out a tax-break strategy for the owner of a potential parking lot, now vacant and adjacent to the theater. So far, the landlord has had little need to develop the lot, but as three new commercial spaces he has developed in that neighborhood are set to open soon, the parking lot will become a reality. The non-profit hopes to convince the landlord that if he offers short-term use of the lot for off-hour theater activities occasionally, he could get tax breaks that would benefit him and reduce his costs. This may be a strategy that works for others in similar situations. It helps that the president of the theater non-profit also has his own insurance agency!