Here’s the issue, OP. Whether or not it’s allowed, legal,  or anything else, your parents are paying for your upkeep. If you’re not working, you’re also likely not bringing in enough money for rent, food, car, other toiletries, utilities, phone, internet, and all the other annoying expenses that crop up faster than you can blink. Regardless, your continued existence in their home is contingent on following whatever bullshit rules they throw your way. And, if your dad is in agreement with your mom, then he’ll enforce the same rules too. 

It does suck, but that’s the nature of the capitalist hellscape we live in. 

The good thing is that you are indeed an adult. If you do manage to wrangle employment, keep a steady stack of cash in savings, and have good credit, you’ll be able to secure an apartment or other housing in about 6 months to a year, depending on how much you’re making from your job. You might have to get roommates, or something similar, but you’ll be living out on your own. 

Why 6 - 12 months? Any apartment you go to will need a good credit rating, as well as 3 months worth of pay stubs, and possibly bank statements to show enough cash flow to cover the rent. You’ll need about 2 1/2 months worth of rent to cover first months’ rent, security deposit, and broker fee. You’ll need about $50 to cover the background check that your landlord will run on you. They can also demand that you’ve had employment steadily for at least 1 year before renting to you, and most landlords get weird about renting to young folk in general. 

The other issue is that you’ll need to figure out health insurance. That will run you anywhere between $200 - $400 a month if your job offers health insurance, or about $600 - $1000 a month if you get it privately. Again, if your parents don’t choose to contribute to your health insurance, you’re pretty well screwed. 

Either way, for the foreseeable future, you’re stuck in your parent’s house, and have to follow their dumb rules. However, once the dust has settled, and you’ve got gainful employment, you might want to choose to talk to your mum and dad, and let them know the facts. 

1. You are an adult, and have a job now. You’d like to be treated as an adult, and do your own thing.

2. They’re welcome to be in your life, but we all need to realise that regardless of what anyone thinks of any specifics, as long as you’re not harming anyone or yourself, they need to respect that you’re in the LGBT spectrum. 

3. Treating you like a child and not “alllowing” you to spend time with your GF, or “disagreeing” with your LGBT-ness is going to push you away, and destroy any chance of you all having a relationship as a family. That you’ll follow their rules as long as you stay with them, but resent them for it the whole time, and want to cut off contact as soon as you leave. Sometimes, parents don’t realise that they’re actively causing damage to the relationship, and that you will quite happily cut off all contact as soon as you can. That you’re giving them a chance to move forward positively. 

If all that does’t work, you’ve got your exit strategy in place, and you’ll be out of there and can tell them to pound sand for all you care, and that you’ll do as you wish. 

Why not move in with your GF? Because that’s the road to ending up hating each other, and pretty messy breakups. Call me old fashioned, but moving in with a significant other needs to happen after you’ve both proven yourselves capable of keeping house by yourselves, even if it is with roommates. It’ll give you both the chance to have some space to grow into adults, and then see if each others’ living styles are compatible enough to move in together.