  Nearly quantum limited Josephson parametric amplifiers (JPAs) are essential
components in superconducting quantum circuits. However, higher order
nonlinearities of the Josephson cosine potential are known to cause gain
compression, therefore limiting scalability. In an effort to reduce the fourth
order, or Kerr nonlinearity, we realize a parametric amplifier with an Al-InAs
superconductor-semiconductor hybrid Josephson junction (JJ). We extract the
Kerr nonlinearity of the Al-InAs JJ from two different devices and show that it
is three orders of magnitude lower compared to an Al-$\text{AlO}_\text{X}$
junction with identical Josephson inductance. We then demonstrate a
four-wave-mixing (4WM) parametric amplifier made with an Al-InAs junction that
achieves more than 20 dB of gain and -119 dBm of compression power, that
outperforms single resonant JPAs based on Al junctions.
