My comments below.
1a. You say you try to move the outside ski over to match the inside ski. But but but - the ski that can be moved is the one you are not standing on. Sounds like you are standing on the inside ski. But to ski properly, you need to be standing on the outside ski. It's the one that turns you. Somehow you have this backwards. Putting your weight (standing) on the outside ski applies whether you are in a wedge or parallel. So... the ski you need to move over to match is the inside ski. It should be light.
1b. If you are having trouble moving the inside ski over, you have too much weight on it. But when a skier is skiing in a wedge, both skis hold much of the weight. That's the right way to do a wedge turn. So the weight distribution has to change to get parallel. In order to shift from a wedge to parallel, you need to teach your body to stand on that outside ski with almost or all of your weight on it, and learn to do this at least long enough to slide the inside ski over to match.
-- --Thus the tapping exercise. It's the uphill ski (inside ski) that needs to be lightened and moved over. Focusing on tapping it often prompts a skier to stand on the other, downhill, ski. Unconsciously. If that isn't working for you, focus consciously on standing on the downhill ski and sliiiiide that uphill ski tail over so both skis are pointing in the same direction.
1c. This tapping exercise works best when learning if the skier tries to go across a slope and tap the uphill ski. No turning, just go across. Then turn at the end and go in the other direction, tapping tapping tapping. Over and over in both directions until you can do it. Unacknowledged fear of falling on your side downhill when you stand on the outside ski can make this task quite difficult.
1d. One way to overcome the fear of falling is to bounce- bounce-bounce on the outside ski, as you make wedge turns. It will be the downhill ski for much of your turn, so you can think of this task as bouncing on the downhill ski if that's easier and less confusing. You can do this when linking wedge turns; no need to go across the slope to learn it. When you bounce on that outside/downhill ski, the bouncing puts more weight on it. Bouncy-bouncy! Do this when making wedge turns, over and over. It's fun. Then, when it's fun and you're laughing, try sliding that inside/uphill ski over while you're bouncing. I bet you won't be overcome with fear of falling.
2. You say when you take your mind off holding the skis parallel you lose the parallel stance. Welcome to the process of building ski skills! There are SO MANY DISTRACTIONS in skiing. Too many to keep your focus on one thing for a whole run. The old habit will return as soon as you're no longer focusing on doing the new movement. This will be the case as long as you continue to build skiing skills. SO.... holding the focus, while destined to be a real and constant issue, is your challenge. Strength has nothing to do with this. Just recognize that you forgot to hold the skis together, forgive yourself because it couldn't be helped, congratulate yourself for recognizing that the old habit of wedge stance has returned with out permission, and push the reset button. There's no other way.
3. You have been told the permanent weakness is minor, and shouldn't affect normal life. So it shouldn't affect skiing either. Every skier has a weak side. It will always need work to get it to behave like the strong side. Focusing on getting that weak side to act like the strong side will produce good results if you stick with it. You don't need to be stronger. You need to work on holding onto that focus. Eventually the old habits on the weak side will be overwritten by new better habits. But when you get into situations where you are genuinely frightened, the old stuff will return. This is normal. The longer you ski, the less often this will happen.
4. You are trying to learn to tap while doing full turns. That is a difficult and confusing way to learn. You may be getting mixed up on which ski to tap. Try it in a traverse (look uphill and don't go until the coast is clear), and tap the uphill ski only. Do it in both directions. Try the bouncing as described above it your body refuses despite being asked politely.
5. It sounds like you are trying sideslips while the skis are in a wedge. Is that right? It won't work. In a wedge, the uphill ski's Big Toe Edge will grip the snow, no matter who is on the skis. Try side-slipping with skis parallel. All you're doing is going straight down the hill. Keep both skis on their uphill edges. Use your ankles and knees to dig those uphill edges into the snow so you stop, the use you ankles and knees to lessen the grip by slightly flattening them. You'll need to be on a bit of hill that's got more pitch than the flattest part of the bunny slope or you won't go downhill at all. Find a little spot that's short and looks a wee little bit steep, and try it. Work on this facing both directions. If your skis take off, you've got them pointed in the wrong direction. Adjust by trial and error until they don't want to go forward or backwards. DO NOT use the wedge to get control of this. Re-point both skis while parallel to a new angle to stop the forward or backward movement. It's just like side-stepping uphill. If you have trouble adjusting the point of the skis, try side-stepping up the little steepish part and adjust the way they point, keeping them parallel, to teach your body which way to adjust them when they go forwards or backwards. Repeat until it's easy. DO NOT use a WEDGE to control this. Your skis will take off on you. That's because one of them will want to go downhill and believe me, it will. So will you.

Parallel is the way to sideslip.
Doing the tapping and the sideslipping will help tremendously in getting you able to ski parallel. It takes some people years to get parallel, if they don't ski frequently. Be patient. You can do this.
Did you know skiing would ask so much of your brain???? It's not strength at this point in most people's skiing that builds new skills, it's the brain (focus, and focusing on the right things) that gets things to work.
Good luck on getting parallel!