Parking infrastructure is pervasive and occupies large swaths of land in cities. However, on-demand (OD) mobility has started reducing parking needs in urban areas around the world. This trend is expected to grow significantly with the advent of autonomous driving, which might render on-demand mobility predominant. Recent studies have started looking at expected parking reductions with on-demand mobility, but a systematic framework is still lacking. In this paper, we apply a data-driven methodology based on shareability networks to address what we call the “minimum parking” problem: what is the minimum parking infrastructure needed in a city for given on-demand mobility needs? While solving the problem, we also identify a critical tradeoff between two public policy goals: less parking means increased vehicle travel from deadheading between trips. By applying our methodology to the city of Singapore we discover that parking infrastructure reduction of up to 86% is possible, but at the expense of a 24% increase in traffic measured as vehicle kilometers travelled (VKT). However, a more modest 57% reduction in parking is achievable with only a 1.3% increase in VKT. We find that the tradeoff between parking and traffic obeys an inverse exponential law which is invariant with the size of the vehicle fleet. Finally, we analyze parking requirements due to passenger pick-ups and show that increasing convenience produces a substantial increase in parking for passenger pickup/dropoff. The above findings can inform policy-makers, mobility operators, and society at large on the tradeoffs required in the transition towards pervasive on-demand mobility.