RC Debounce: Is there a benefit to placing the capacitor in parallel with the switch?
Is there any disadvantage to tying the capacitor to the opposite logic
level as the switch? Like this:

Yes. Now the switch is connected to the supply voltage, not ground. In some cases this may be a problem, eg. possibility of cable short from supply to ground, higher EMI due to switch not grounded, static discharge could be injected directly into supply, switch might share common connection with other parts that must go to ground.
The other alternative is to wire the capacitor to the high side. This is perfectly fine.