How to control one Mosfet with another mosfet
There's one thing you missed with your arrangement, and that is getting the 12V to turn the MOSFET on.  In your current arrangement it's possible to turn it off, but never on.  Adding a pullup resistor can help:







simulate this circuit &ndash; Schematic created using CircuitLab

R1 pulls the gate of M1 up above the switching threshold to turn it on. M2, when on, pulls the gate down below the switching threshold, turning it off.

However: you may have noticed that the switching is now backwards: send a HIGH from the Arduino to turn it off. Which of course has the effect that the output is normally on and you have to actively turn it off.

Again a pullup resistor can help:







simulate this circuit

Now R2 keeps the gate of M2 pulled up (switched on) which keeps M1 turned of. That is, until you actively pull the gate down to ground by sending a LOW from the Arduino.

To ensure that the peltier never turns on when you don't want it to you should also take steps to ensure that the 12V power is never provided without the 5V also being provided - so you could maybe power the Arduino through the barrel jack from the 12V supply.  It doesn't matter if the 5V is provided but not the 12V, but the other way around could mean the peltier comes on with no active control over it.