  We demonstrate fast two-qubit gates using a parity-violated superconducting
qubit consisting of a capacitively-shunted asymmetric Josephson-junction loop
under a finite magnetic flux bias. The second-order nonlinearity manifesting in
the qubit enables the interaction with a neighboring single-junction transmon
qubit via first-order inter-qubit sideband transitions with Rabi frequencies up
to 30~MHz. Simultaneously, the unwanted static longitudinal~(ZZ) interaction is
eliminated with ac Stark shifts induced by a continuous microwave drive
near-resonant to the sideband transitions. The average fidelities of the
two-qubit gates are evaluated with randomized benchmarking as 0.967, 0.951,
0.956 for CZ, iSWAP and SWAP gates, respectively.
