Is she overreacting? Probably not.
MANY people are mildly allergic to cats. They just get miserable; eyes swell shut, difficulty breathing but still can breathe, shuts down their life for a few hours to a few days. For some people, however, the allergy is life threatening.
If your mother and step father are notoriously controlling / wage battles for your affection over your father... then they MAY be manipulating the situation. But typically, ****people with allergies DON'T overreact****. If they're deathly allergic: They're justifiable terrified. It's like a toddler with a loaded gun running in the house (would you invite one over?). Sure, they might live this time, but next time??? If they're mildly allergic it's like having someone who is *actively* throwing up with the stomach flu come over. Neither situation is one any sane person willing invites into their home. I mean, would you want your friends to bring a child to your home who was virulently ill? ((I can just imagine the MP Q: Friend brings over toddler who is throwing up, and says "It's just the stomach flu, I know you'll get it, but it's just a few days of your life, what's the big deal??"... Is that rude? would get an OVERWHELMING "That self centered, uncaring slitch, how dare she intentionally make you sick?! How RUDE!" )) But for some reason, people blow off allergies, even though they're as bad as the flu for that poor person, IF they're not lifethreatening... And that's like coming over and shooting the person.
Seriously, though, here's my fun time with life threatening allergies (JFF), I'm sure you understand life/death situ, but it's another thing to really witness someone's struggle to stay alive because other people say "it's no big deal" to a major allergy.
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An exboyfriend of mine was so allergic to cats that he had to epi pen himself a few times a week when he was in a populated area just from being in the same room with someone who had been in the same room with a cat a few days prior. If he breathed by them, his throat would start to swell shut. If they actually owned a cat his lungs would shut down. If he was touched by a person who had touched either cat OR a person who owned a cat his skin would blister and then he'd go into full anaphylaxis. We once went to a friend's house (didn't own cats) and left via ambulance because the previous tenants (house had been thoroughly cleaned by professionals, and was ongoingly cleaned by our friends) had had cats. He was AS allergic to cats as some people are to peanut butter. But peanut butter is pretty visible and fairly easy to clean up. Cat dander is microscopic. He was allergic to both dander AND the oils that cats produce. He was mildly allergic to many kinds of animals, but he was deathly allergic to cats.
Most of the time his epipen would work as long as he removed himself from the situation (he also kept a 5gal jug and clean clothes in a bag in the truck. He'd jab himself, strip naked, seal his clothes in a ziplock bag, pour the water over himself, scrub, dry, and change into new clothes. If he kept reacting, but mildly, we'd go to the ER. If he kept reacting strongly we'd call the EMTs so they could shock and intubate him if necessary (only necessary twice in 3 years, but we were in the ER so often that we got on a first name basis with half the staff).
My exboyfriend had to RADICALLY alter his life from most people to prevent anaphylaxis. He could never take public transportation, or attend parties with people we didn't know. He spent most of his time out of doors (he was a fishing guide). He had health insurance premiums that were through the flipping roof. Each and every single time he was exposed to cats or their oils/dander his reaction would get stronger. When he was a kid, it was fairly mild. By the time he was a teen his throat would swell up. He had to quit highschool & get his GED the 2nd time he went into full on anaphylactic shock. He was saving up for moving completely off the grid out in the middle of nowhere (but on a medevac path, with chopper landing site), and was nearly there (only another 50k to go) the last time I spoke with him.
I was SHOCKED at the sheer number of people who would just "blow him off" ((One time he had to get intubated the person "put the cat in another room, didn't think you'd notice)), or say he was just fond of attention/ drama/ <rolls eyes>. He was a VERY outgoing person, but people called him shy/ antisocial/ etc. because he very very rarely would go out, and NEVER had people over to our home (we couldn't decontam it if someone brought dander over with them).
It sounds like your father isn't as allergic to cats as my ex, but even if it's "just" flu like symptoms... would you want someone bringing the flu to your house?