IRFZ44N getting very hot in Peltier driving circuit
Since no MOSFET drivers are shown on the schematic, I assume the FETs are driven directly from arduino pins. IRFZ44 requires more than 5V Vgs to turn on fully. With 5V gate drive  it will be somewhere in the linear region. So you must either use a FET with logic level gate drive, and a proper MOSFFET driver with a 5V supply, or keep your FETs and use a proper MOSFET driver with a +12V supply.
Likely explanation for the uneven heating:
FET RdsON has a positive tempco so FETs in parallel share current well. Resistance in the hottest one increases which directs current to the other FETs.
However in linear region it's the opposite. Threshold voltage has negative tempco, so it goes down as it gets hot:

This means your FETs run in conditions that are ideal for thermal runaway.
Additionally, the absence of gate resistors mean they are likely to oscillate due to layout parasitics ; and 1MHz is way too high a frequency. You must calculate inductor ripple current according to frequency and do the math to design a buck converter.
With a proper FET driver chip using 12V power supply you should be able to drive these FETs at 100kHz without trouble.