Go talk to your doctor about getting a prescription for an allergy medication. Also check about setting up an appointment to see an allergist. Since you want to have the cat come live with you, you are going to be willing to do all this, but unless your parents are also willing to go into the doctor, see an allergist, get and take an allergy medicine, it's all for nought. Make sure that they would be willing to have the cat enter your life too.
If you do wind up bringing the cat to live with you, it will be a bit of a slow and gradual process in introducing her to the dogs. Some dogs are extremely gentle around cats, others are incredibly curious, some just want to play. The cat also needs to have a behavior and demeanor that will meld with your dogs. Have you seen the cat interact with dogs before? Is the cat a little skittish around new people, things, moving objects? Is the cat maybe a little assertive? You won't be able to get a true understanding of how they will react to each other until they finally meet each other.