We demonstrate an optical parametric chirped-pulse amplifier producing infrared 20 fs (3-optical-cycle) pulses with a stable carrier-envelope phase. The amplifier is seeded with self-phase-stabilized pulses obtained by optical rectification of the output of an ultrabroadband Ti:sapphire oscillator. Energies of -80 microJ with a well-suppressed background of parametric superfluorescence and up to 400 microJ with a superfluorescence background are obtained from a two-stage parametric amplifier based on periodically poled LiNbO3 and LiTaO3 crystals. The parametric amplifier is pumped by an optically synchronized 1 kHz, 30 ps, 1053 nm Nd:YLF amplifier seeded by the same Ti:sapphire oscillator.