The potential rapid growth of paid ridesharing services brings a marketplace to utilize the empty seats in commuting vehicles. Yet few studies of morning commute have included paid ridesharing in their analysis. This research formulates a continuous-time dynamic ridesharing problem for a single bottleneck in the morning commute. Travelers' choice of departure-time and ridesharing mode as groups with heterogeneous values of travel time. Parking is introduced in our analysis for system optimum from the perspective of the system management, where the parking charge is shared among the driver and passengers in a same vehicle. Dynamic parking pricing strategies to achieve the system optimum with no queues at the bottleneck is then derived. The morning commute problem is then converted into a differential complementarity system (DCS), so that the discretized problem can be solved numerically. It is found that in the ridesharing scenario, the travel time can be a piecewise linear function for each early and late arrival time segment of every heterogeneous group, and the corresponding demand rate is a piecewise step function for each group. Such performance is much more complicated, compared to the linear travel time function and constant demand rate for each arrival time segment in solo driver scenario in the literature. The analysis and numerical results further show that under different ridesharing payment policies, the system performances, such as group-specific costs, vehicle-miles-traveled, vehicle-hours-traveled, total costs, would be quite different, which suggests that the ridesharing payment policies should be properly designed to achieve the social, economical and environmental goals.