Yes, you can connect the switches in parallel.
They all need to be the type that's open circuit when the door is closed. So if it's a pushbutton type switch, which is held in the "pressed" position while the door is closed, it needs to be a "normally closed" pushbutton, because "normally" refers to the state when nothing is pressing the button. A normally closed pushbutton will be open circuit when it's pressed, i.e. when the door is properly closed.
If you connect two or more of these switches in parallel, there will be continuity when any one (or more) of the switches are closed, i.e. when any one or more of the doors are open.
Then you need a power supply and an alert of some kind. You won't need a relay unless the switches have a very low current rating and/or the alert needs a lot of current or a high voltage. A typical microswitch can switch 1A at 120VAC or 28VDC which should be enough for most situations.
If you use reed switches instead of microswitches, you'll also need normally closed types, which open when the magnet is nearby.