From the picture shown, I see three things that can help:
1: Shock mounts like Shadow Catcher mentions. That way, vibrations are not transferred from the pump to the rest of the rig which is acting like the sounding board in a guitar, amplifying the noise. Dealing with vibrations on the cables and hoses also helps.
2: A small accumulator tank, like the ShurFlo. It doesn't store much water... but it is pressurized to 30 PSI, and helps keep the brief "burrup" noises at bay.
3: Worst case, get a variable speed pump.