Mixed usage of radio frequency (RF) and free-space optical (FSO) transmissions can improve the reliability and throughput of future satellite communication (Satcom) systems. To overcome the rate limitation of conventional hybrid implementations, we propose parallel FSO and RF transmissions to explore their complementary properties in beamwidth and bandwidth. In particular, RF transmissions serve users over a large geographical area, while the FSO link is employed to increase throughput to a particular hotspot area with higher capacity demand through an access point. Independent data streams are adaptively sent over both links to satisfy capacity and availability requirements. To highlight the significant potential of parallel FSO-RF transmissions, we analytically derive the sum capacity outage probability. The obtained expression is validated by Monte Carlo simulations. Furthermore, we derive an asymptotic expression to obtain diversity gain and provide further insights into the behavior of the proposed system. The presented results demonstrate the potential of parallel FSO and RF Satcom over existing solutions.