Need advice for introducing cats

I have never introduced cats before - my last cat passed away a year ago, I had her since I was 5 and she was the only cat I knew. Shortly after she passed, I adopted 2 cats and luckily they got along almost immediately, it helped that one was a kitten.
Unfortunately I've had to move back in with my parents, who have their own cat (they just adopted her 2 months ago). I may be here for a while so it's important that they get along.
The resident kitty is a 1.5 year old female. I have a 2 year old female, and my other one is a 1 year old male. They are all spayed/neutered.

It's been 5 days now and this is how it's gone:
The first 2 days I kept them in my room, where they have their litterbox, food/water and cat trees.
For the last 3 days I would close Resident kitty's door and let mine explore the house, for a few hours.
Yesterday, I tried putting mine in a kennel so Resident kitty could inspect them, but mine get very upset being in a cage, and Resident kitty doesn't want to get near them, she stays a good distance away and hisses/growls. Mine don't reciprocate, they just seem very curious about her.

So, today I decided to leave Resident kitty in her room and put a gate at the door so that she can watch from afar. She still hisses and growls if they come to the gate, but she seems comfortable and isn't hiding under the bed like she was. I figure I will do this for a while everyday, and hopefully she will eventually take some interest in my cats and want to check them out.
I know everything online says I should do it the other way around, but Resident kitty is very shy and still pretty new to the household, so I think it's best to leave her in the room that she's comfortable in.

Does this seem like a good plan? Is there anything else I can be doing to help this process?

I think you may be going too fast too soon. Site swapping is fine, but it sounds like your parent's cat just isn't ready for visual contact. Scent swapping is the next step - wipe each cat down with a clean washcloth, focusing on the scent glands on the cheeks. Really saturate the cloths with one cat's scent (don't mix them up). Then leave the 2 cloths from your cats where your parents cat can find it and vice versa. Repeat each day with clean cloths. Don't move on to visuals until each cat shows no negative reaction to the other's scents. Site swapping can continue throughout this process. Only when all cats are ready do you move on to visuals - start feeding either side of the baby gate (safer is 2 gates stacked on to fill the doorway, or a screen door), but make sure you have a blanket or something over it - to start with you want scent but no visual. As the cat's get confident you can gradually raise the blanket a little each day, until they have full visual contact. Once that is all going well, they can have supervised meetings, and eventually full integration.

Interactive play is really important for all three cats during this process - you want to drain their energy constructively to minimise the chance of any fights. Interactive play is also a great way to build self confidence and confidence in the territory.

These articles may help :-
http://www.thecatsite.com/a/the-ultimate-yet-simplified-guide-to-introducing-cats
http://www.thecatsite.com/a/introducing-cats-to-cats
http://www.thecatsite.com/a/the-multi-cat-household
http://www.thecatsite.com/a/playing-with-your-cat-ten-things-every-cat-owner-needs-to-know
http://www.thecatsite.com/a/cat-play-the-rules-of-the-game

Feliway diffusers throughout can help ease any stresses that arise during this time.

I will definitely focus more on the scent-swapping. I would have taken things slower but my parents have made it difficult because they're impatient, they keep saying I should just let them out and let them get used to each other. And they let them out sometimes when I'm not here. But I know that's not a good idea as long as their kitty is still scared of mine, I want her to feel totally safe first.

I hear you on the parents thing - I'm an adult living at home too. You may find that if you show them the articles I posted links for, and explain why it's better to go slowly they'll get a better understanding and come on board with the process.

I'm juggling new cat intros with parents more old fashioned views too - there have been a few tempers flaring along the way, but they do now understand and support the process.

Good luck