The simplest description that’s still accurate of the Twin Paradox is that you can use any reference frame you want for the calculations, but there are not two relevant reference frames, but three.
Let’s say that, instead of building their own spaceship, Bob hitchhikes, while his twin Alice stays on Earth. Going out, he thumbs a ride with ET, and then when he’s gone far enough, he gets out and ET keeps on going. Then, on the way back, he hails Alf, who’s coming this way anyway, and rides back with him.
You can work the problem in Alice’s reference frame, and in that reference frame, Bob is always moving and Alice is always stationary, so Bob always ages slower.
You can work the problem in ET’s reference frame, and in that reference frame, Alice is always moving, and Bob spends part of his time at rest, and during that period, Alice is actually aging slower. But then for the second leg of the trip, Alice is still moving, but Bob is moving much faster, to catch up to Alice, and so Bob is aging much slower. It’s enough of a difference that, by the time Bob catches up to Alice, he’s still younger than her, by the same amount as when you did the calculation from Alice’s frame.
Or, of course, you can work the problem from Alf’s reference frame. Again, in this reference frame, Alice is always moving, and Bob spends part of the time at rest and part moving very quickly, except now, Bob’s quick-moving part is first and then his at-rest part. Again, you’ll find at the end that the age difference is the same as in the first reference frame.
Or, you can, I dunno, use Zaphod Beeblebrox’s reference frame, who’s going in a completely different direction at a different speed and never gets anywhere near the Earth. I don’t know why you’d use his reference frame, and in that frame, the calculations are all a lot more annoying, but you still get the same final answer once the twins meet up again.
But what you can’t do is use Bob’s reference frame, because Bob doesn’t have a reference frame, he has two reference frames. Sometimes Bob is in ET’s reference frame, and sometimes he’s in Alf’s.
Also of note, by the way: At times before the twins re-unite, different reference frames will disagree about which one is older “right now”. This is fine, because different reference frames will also differ about just what “right now” means.