I'm going to go ahead and say that YES, you can do as you propose, wiring the two thermostats in parallel.
While, as others have correctly pointed out, this will not let you achieve temperature control in both the upstairs and downstairs simultaneously, I'm going to take you at your word that you simply do not want it to be uncomfortably cold in the basement when occupying the basement, and infer that you don't care what happens upstairs during those times.
In this case, go for it--install the second thermostat.
The upstairs area of the house will in fact be over-heated when you use this basement thermostat, but you may still consider the result to be an improvement over the current situation. Just remember to reset the basement thermostat to cooler temperature before you leave to go back upstairs or you will find yourself having to make the trip back down later.
The "correct" way to solve the problem is to split your system into two zones, which requires installing baffles/valves on your heating ducts/pipes, which would allow independent control of upstairs and downstairs.
If you are using traditional thermostats, then adding a thermostat in parallel will change the heat anticipator function in both of them. This is a function that reduces furnace run times to improve temperature stability. If both are always connected, lower the HA settings in each to fix.
In theory this should be easy -- have the second thermostat connect the R and G wires when it calls for cooling, and the fan will run.
Here's one reason it might NOT work, and it's subtle/non-intuitive so I thought it was worth mentioning: Some thermostats short the Y (cooling) and G (fan) wires together whenever they're in cooling mode. They do this so that they can use one relay to turn on both fan and A/C during a call for cooling. (Relays are relatively expensive components.) This shorting happens whenever the thermostat is in "cool" mode, whether it's actually calling for cooling or not. (Although if you set such a thermostat's Fan switch to "ON" instead of "AUTO", the short is removed so as to run the fan only.)
Where there's only one thermostat in the system, this shorting behavior does no harm. But if a second thermostat is added and it calls for "fan", the power will go through the short in the first thermostat and turn on the A/C as well. I have this exact arrangement in the house I just bought, and I cannot run the fan independently from either thermostat -- it will always run the A/C as well. If I unwire either thermostat (or set it to "heat" mode), I can then run the fan without the A/C.
So you might try shorting R (or Rc) and G with your existing setup to make sure that the A/C does not come on as well.
Source: personal observation confirmed by isolated testing on Robertshaw 9600 programmable thermostat
Best Answer
No, the stat needs 24vac for the wifi, which is supplied by the furnace from the R and C terminals. You can easily find on Amazon or other retail sites a "24vac C wire adapter" and you can install the thermostat anywhere near a 120v receptacle.