The increasing availability and adoption of shared vehicles as an alternative to personally-owned cars presents ample opportunities for achieving more efficient transportation in cities. With private cars spending on the average over 95\% of the time parked, one of the possible benefits of shared mobility is the reduced need for parking space. While widely discussed, a systematic quantification of these benefits as a function of mobility demand and sharing models is still mostly lacking in the literature. As a first step in this direction, this paper focuses on a type of private mobility which, although specific, is a major contributor to traffic congestion and parking needs, namely, home-work commuting. We develop a data-driven methodology for estimating commuter parking needs in different shared mobility models, including a model where self-driving vehicles are used to partially compensate flow imbalance typical of commuting, and further reduce parking infrastructure at the expense of increased traveled kilometers. We consider the city of Singapore as a case study, and produce very encouraging results showing that the gradual transition to shared mobility models will bring tangible reductions in parking infrastructure. In the future-looking, self-driving vehicle scenario, our analysis suggests that up to 50\% reduction in parking needs can be achieved at the expense of increasing total traveled kilometers of less than 2\%.