@EdithP, here's what I see, as it relates to what you've posted. Your comments in red, mine in black.
Before the operation on snow vs after the operation on the carpet
--hip operation, then the mandatory rehabilitation
--I had to start from scratch and am now relearning parallel turns.
--In spite of my best efforts, the A-frame is still there.
--when I was skiing on snow, ...I was not finding it as hard as I do now.
Your skiing on the carpet shows important improvements, despite the wedge (A-frame) you are using. It's time to celebrate.
LEANING IN on snow
On the snow, you extend (lengthen, straighten) your new outside leg to start a turn. As you do this, the skis begin to turn to point downhill. At the same time you lean your whole body towards the inside of the new turn. This whole body lean does two things, one good and one bad. It edges both skis, so their built-in turning function starts to work; they bend and turn you. It also positions your upper body, where most of your body weight is, over the new inside ski. That weight needs to be over the new outside ski. When the inside ski has too much weight on it, there will not be enough weight on the outside ski and it will slip downhill as you turn. You'll lose your grip.
UPRIGHT TORSO on the carpet:
On the carpet, you do not let your body "lean in" to edge the skis. In the lext-to-last video, you even lean your body out over the outside ski. This is great! Once you figure out how to easily get your weight, all of it, on that outside ski, you'll be able to loosen that inside ski and bring it over to match the outside ski and the wedge and A-frame will be gone. Leaning out over that ski is the beginning. Flexing your ankles to press your shin into the tongue of the boot is a big part of that as well. Look forward to your instructor asking you to lighten that inside ski at some point in your future so you can slide it over to match the outside ski. When you can transfer your weight to the new outside ski before the turn starts, you'll be able to keep both skis parallel. And you will no longer be leaning in, causing your outside ski to lose grip. So yes, the A-frame is still there, but you are learning how to put your weight on the outside ski so you can get it gone. Be patient. You're on your way to parallel. Don't rush it.
UPPER BODY ROTATION
On the snow, your turn your whole body to face the way the skis are pointing with each turn. You are using a bit of "upper body rotation" to encourage the skis to rotate across the snow. It works, but it's something beginners need to purge from their skiing. On the carpet, you face forward. You are getting your skis to turn without rotating your body along with them. This is an excellent advancement in your skiing. Pat self on back! And give that instructor a big tip every time you have a lesson. He deserves it.
It's harder now, on the carpet, for all the reasons people have said about carpets. But it's also harder because you are leaning some very important movement patterns that are unfamiliar to you. You are working very hard, paying intense attention to doing new things with your feet and legs, so that you make your turns happen by doing stuff with feet and legs, not with upper body leaning and rotating. This is extremely important to learn, and it take intense concentration to do. This is obvious in your rigid stance; you are concentrating hard on doing those things your instructor is telling you to do, and you're doing them! Expect to get exhausted. Good skiers ski from the feet up. You are learning to do just that. Bravo!