AAV's can not be used on washing machines
You can only wet vent bath room fixtures or a bathroom group.
And let's stop screwing around here. You can not, drop a waste line (and that's what the softener and filter lines are) into a vent stack (and that's what that pvc pipe is) under any conditions other than fixtures (sinks) being dumped into a stack vent system and there are very specific codes and conditions that need to be met to do that and nothing there meets those conditions. Plain and simple the installation is not to code and it is wrong and whether or not it works is immaterial. When you start listening to and considering the advice of people that are not licensed plumbers and who would rather fluff their own ego's in an attempt to appear helpful you are setting yourself up for problems. Call a licensed plumber in and have the drain done the right way. An AAV here is not a solution to anything.
I am a licensed master plumber. Have been for over 35 years now. I teach plumbing and I have been a plumbing board member as well as a member of the code review board. I'm not telling you the piping is wrong just to be a pain in the ass and I have no personal stake in your situation one way or the other but it really frosts my nads when unlicensed folks give their uneducated advice on a topic they know nothing about. You don't see me on the medical forums giving half assed diagnoses to people just to fluff my ego.
Baill, the best advice you gave there was to call the local inspector and ask him. That's very helpful and a damn good idea