False. This is nothing but something somebody made up and which has no basis in reality -same as the height dominance theory and the acclimatization of a tropical bird to a cold climate - all made up bunk.
With the exception of quakers, parrots don't live in 'homes' and, although they do have a territory, they don't defend it (they don't belong to territorial species and are not predators) so, even though some parrots would bite if you put your hand in their cage, they are not doing it because they 'own' it or the room where the cage is and want to defend it from others. There are always other reasons... the bird could be overly hormonal, it could be nesting during breeding season, it could simply not trust the human or it could be what I call 'cagelized' (like convicts who become 'institutionalized' and only feel safe in jail).
I open my birds cages and allow them to come out on their own and this is something that I do on purpose to the point that when I get a bird that only comes out if you make it step up, I actually wait for as long as it's necessary (and this means months sometimes) for the bird to do it on its own. The reason for this is that I always try to emulate their natural way of living as much as possible and part of that is allowing them the right to decide. I believe they are happier and better adjusted when they are self-sufficient and making as many decisions as possible on their own (the way it would be in the wild). I have a sun conure that would not come out on her own and it took months for her to have the nerve to do it and more months until she started flying around on her own. She had not been taken care of in the right way but not because her owner did not love her or meant to do anything wrong - he simply did not know any better. She had a real bad diet (so bad that she ended up with hemochromatosis from it) and was very insecure because she had been kept in a laundry room all by herself and taken out very little, with the consequence that, although she was not clipped, she could hardly fly (clumsy and weak) and only did it when she was startled into it (she is doing great now, comes out of her cage as soon as I open it, flies around, is eating a good diet, etc -she is laying on my chest as I type this, grinding her beak nonstop
).
And I also don't think that parrots hold grudges... not for long anyway. In my personal experience, they are actually incredibly forgiving animals and very, very patient with our mistakes and the hardships of captivity but they also don't wait forever... eventually (and this could take years), if you are not giving them what they need, they will show it one way or another (screams, bites, plucking, self-mutilating, etc). I also think that birdrooms are wonderful but ONLY if you have a number of birds that gets along in them, and ONLY if it means a room that is meant for birds and for nothing else so the birds can live cage-free in safety. A birdroom is not a human room where the bird's cage is physically located. My birdroom has cages in it but there is nothing in them unless I need to put one in it for whatever reason (they use them as 'stands' and to climb up to the platforms). Right now, the male/female amazon pair is in a double macaw cage when I am not in the birdroom or if people besides me are going to be going in but that is only because it's breeding season and they have a nest with eggs (their second clutch) and the male gets very protective and thinks nothing of attacking when he feels somebody or somebirdy is getting too close to his mate or the nest - and he can do A LOT of damage with his beak and claws! But, once they abandon the eggs/nest and start the molt, they will be allowed to live cage-free all the time, again.
I noticed that you have Zoe's cage in the laundry room and I think that's a mistake and part of the reason why she is now biting you more and more often. Parrots only want to be by themselves when they are sick or wounded, otherwise, they need to have constant company. Being in a room by themselves is terribly stressful and depressing to them - especially for grays because they are very needy birds and require being right there with their human all the time or they become unhappy with their human and their life, in general. I think her cage should be next to your desk or in the room/space where you spend most of your day and that she should not eat on her own, I think you should eat with her - eating is a social event for them and they always do it in flock, never on their own. My birds are let out of their cage at dawn (at 5:30 am this time of the year and, although I could start doing it a bit earlier because there is already light in the sky by 5:00 am, I 'reserve' the 4:45 or 5:00 am uncovering time for the summer), get their produce at around 6:30 - 7:00 am (I eat some with them) and their gloop at around 7:45 - 8:00 am (they go back into their cage to eat it and come out again at around 9:00 am or so -the times depend on how bright the day is because their schedules don't go by the clock but by the sunlight so gray days are shorter than bright sunny days) and stay out until 1 pm when they go back in for their noon rest. In the summer, when I don't have to take care of my grandkids after school, they would come out again for another 2 or 3 hours in the pm.
Why don't you tell us what her diet, light schedule and daily routines (if she has any) are? Because I don't think that she is biting you because you go away twice a year, I think there is something else going on and that you might need to re-evaluate your husbandry from the room where she has her cage to the light schedule, diet and daily routines (this is another VERY important part of making them happy). I hardly ever go anywhere so, normally, my birds schedules and routines don't change but, every now and then, I need to go somewhere like 'normal' people do and I know for a fact that although he denies it, my husband does not keep the schedules as faithfully as I do (he does not get up early for anything except his job!
). We took a grandkid to Canada (a week) three years ago, two years before that I took another one to Orlando (a week), four or five years ago, I went back home for my niece's wedding (two weeks), we took a son and his family to Washington DC for a couple of days last year, I am going to be taking one of my grandkids to Orlando in October (a week), I will be visiting family back home a few months after that (two weeks) and a couple of years ago, I also had to spend ten days in a hospital taking care of one of my daughters so it's not as if I don't ever disappear on them because I do but I never have a single problem with them - quite the contrary, they are all super happy to see me!