So I want to run a Raspberry Pi 3 off of thermostat wire. My question is, will the thermostat wire supply enough current for the raspberry pi (2.5 amps)?
Check the wire size. Generally when you go up 4 wire sizes ( 8 --> 12 --> 16 or 10 --> 14 --> 18 ) you cut your current capacity in half.
So whatever your thermostat wire size is, subtract 4, get the ampacity from table 315.16, and divide by 2.
Since the operating voltage of a Raspberri Pi is 3-5 volts, your bigger problem is going to be voltage drop. You'll need to run the numbers and see if the voltage drop at that current is acceptable. What is trivial drop at 24V is unacceptable at 3V.
Do not parallel conductors - even when this is allowed, on very large wire, each wire must be fused separately - including neutral/return. However since the failure mode of one cable would be voltage sag, you won't want that.
This thread discusses how to power a Raspberry Pi with 24VAC from HVAC systems