It is as you described

I heard it also may be related to an old tstat. However I have no idea how to tell how old is old.

So hrv is going to r and g
Ac is going to y and c
Make sure there is not a jumper wire between y and g
If that checks out look at the back of the t stat make sure there is no jumper on the t stat from y and g. Check the wiring on the back of the t stat. Note the colors going down to furnace and make sure they match up. Ie w-w
R-r and rc if ther is one y-y and g-g

Make sure there is no jumper from the hrv to the furnace. There shouldn't be but some people's children are idiots

Hrv is going to r and g
AC is going to c and y
Tstat is going to w r y and g and they match the same color as at the tstat
I don't see any jumpers at the furnace or tstat except the r on the tstat to an empty r

That is really odd. Everything sounds correct. Somehow the circuit is connected from the r to y via your hrv. Try disconnecting the t stat wires and swing if the problem still occurs. If so it is in the furnace wiring if not the problem is in the t stat wiring or the t stat itself. The wire from the hrv to the furnace is to make a contact from r to g to force the furnace fan to run when the hrv is on. Write down the wire colors to terminals so you sent chasing your tail when you re connect everything. So you know what your dealing with, the r is the positive feed off the furnace, when it is connected( via tstat) to w furnace calls for heat, g furnace calls for fan y furnace calls for fan and Ac contractor is energized

I have read a lot about an air exchanger back feeding (incl those who help me on this board). It looks like I just need to get an isolation relay to stop the back feed. Unless you think that is incorrect?

Most heat/cool tstat have a build in jumper. so that when the stat is turning on the ac, the furnace fan turn on.

On a heat/cool stat.
RH is for heating, the high limit control the fan in the furnace.
RC is for cooling, the stat turn on the furnace fan and the ac.

You need to add a relay to prevent the ac turning on when you run the HRV.

If your HRV ducts are connected to return air of the furnace then the furnace fan should run when the HRV is on.

exactly, when fan switch is in auto, y and g are jumped, this back feeds to turn on the AC. Common problem all Hvac guys run into.

Why don't you ask your electrican friend to install a relay for you?

I alway turn off the breaker to the ac in the winter.

Yes this would make sence except he said this still happens when the t stat is switched off. Also I don't think most t stats jumper the y and g when in auto. I'm pretty sure the furnace board takes care of this. Or else when the fan switch was in off, and the t stat was in cool, the fan would not run. Sometimes on older furnaces without a y point people will jumper these points but op said thier was no jumper. The r and the rh need to be jumpered out unless you have a system with separate heating and cooling transformers. Which you don't. If the unit does not do this with the t stat wire disconnected at the bored then double check the t stat wiring and replace the t stat. I do not believe a relay is the way to go. I Hooked 2 of these up last week alone. The only way you should need a relay is if it is an older furnace with no y point.