We report on the generation of four spatially multiplexed picosecond 40-GHz pulse trains in a km-long 6-LP multimode optical fiber. The principle of operation is based on the parallel nonlinear compression of initial beat-signals into well separated pulse trains owing to intra-modal multiple four-wave mixings. A series of four 40-GHz dual-frequency beatings at different wavelengths are simultaneously injected into the LP01, LP11, LP02 and LP12 modes of a 1.8-km long graded-index few-mode fiber. The combined effects of Kerr nonlinearity and anomalous chromatic dispersion lead to the simultaneous generation of four spatially multiplexed frequency combs which correspond in the temporal domain to the compression of these beat-signals into picosecond pulses. The temporal profiles of the output pulse-trains demultiplexed from each spatial mode show that well-separated picosecond pulses with negligible pedestals are then generated.