So, you've got your Parallax Single Relay Board. Next you need to connect this to your furnace, fan, and A/C control wires... and, not burn down your house in the process. Got that? Good.
Refer back to your thermostat wires. We need to connect a Relay Board to each of the control wires for the Fan, Heat call, and Cool call. Here's a reminder which wire is which.
Red - R - 24VAC
Red - Rh - 24VAC (dedicated to heat call)
Red - Rc - 24VAC (dedicated to cooling call)
Green - G - Fan on
White - W - Heat call
Yellow - Y - Cool call
Blue or Black - C - Common
You will need 3 Relay Boards connecting to the Raspberry Pi to control the 3 house thermostat control wires (Green for Fan, White for Heat, and Yellow for Cool). First, make sure on your Adafruit PiTFT has the 2x13 male header properly soldered on to it to daisy chain your RPi header pins (to allow your Relay Boards to connect down to the RPi). This way your Adafruit PiTFT can use the SPI pins (SCK, MOSI, MISO, CE0, CE1) plus GPIO #24 and #25 for its use, while we use GPIO #00, #01, and #04 for the Relay Boards. It's nice to share.
On each Relay Board, connect the + to the +5 VDC pin on your Adafruit PiTFT (male header pin #2). Make use of continuous jumper wires to share the one +5 VDC pin with all 3 of the Relay Boards. Then, connect the - to the GND pin (male header pin #6) using a continuous jumper wire to share this pin also. And, finally connect the S pin of one of your Relay Boards to GPIO #00 (pin #3), one S pin to GPIO #01 (pin #5), and the final S pin to GPIO #04 (pin #7).
Then connect all 3 Relay Boards Common screw down connector to the C - Common wire of your thermostat. Connect one Relay Board Normal Open (NO) screw down connector to the Green Fan wire, one NO screw down connector to the White Heat wire, and the final NO screw down connector to the Yellow Cool wire. Cool? Cool!
You are good to go for the next step!