I don't know why everyone's feedback about feet rising or knee tracking or your choice in shoes are leading to your hip pain. If anything, what I see in your video should cause you to have knee pain.
The only thing I can comment on is that when you go down into the hole, you get to a stopping point then push yourself a little further down not in the first couple reps but definitely during your last few reps. It could be that you are already at the limit of your hip socket mobility at your first stopping point, but because you want to go below parallel, you are forcing your hips to go further than your structure allows and impinging something along your hip flexors.
What this tells me is you need to experiment with the width of your stance so you can still go low without squishing something important in the front of your hips.
Another thought could be weak glutes. This is a personal issue of mine and have been doing weighted glute bridges to help my case. The glutes help keep the leg in the socket at the bottom of the hole (for me at least) so a weak glute can contribute to your legs pushing forward out of the hip socket enough that can cause impingement.
A good test is to go and hold a 1 legged glute bridge see how hard it is to maintain it. If you are shaking or your form breaks down relatively fast, your glutes are probably weak.