Should someone in my family be buying a parrot?

Having a bird is no joke and can be very expensive.  They live upwards of40-60 yrs. if well cared for.  Having the other animals does make it tricky because it would be a shame to have the bird stuck in the cage all the time.  These birds are very expensive and a lot of work.  So many times people get them then realize they cannot take care of them and they get returned.  I wouldn't even think about getting one before getting educated about them prior because they need special care.

It would be a terrible idea with three dogs.  One goof - leaving the bird unsupervised and unsecured, and the bird will be dead.

You won't convince them unless you can come up with a reason and you haven't mentioned any in your question. You hint at the dogs and cat being a danger to the bird. Personally, I'd bet on the parrot in a fight.

u have a cat and they hate birds and will probably kill it if given the chance

Animals in isolation in cages away from their natural habitat is never a good idea

First things first. Arrange for the whole family to go to the pet store and see the birds themselves. If everyone reaches home without anyone sneezing then you're in trouble. Definitely no allergies card to play. Your only hope of argument left is for the two dogs and the cat terrorizing the poor bird. That young bernie and that springer are always on the go. Add a cat that may potentially be Sylvester and you pretty much get an episode of Looney Tunes every single hour of the day.

Personally I oppose owning any parrot. Many of them are stolen from the wild, causing wild populations to drop. Aside from that, African grays have very long lives, and bond tightly with owners. Unstable households shouldn't get one at all.

If you want something from Africa why not get a rhinosauras. They wont fly away if the get out of their cage.