Almost everybody will favour a certain side of the body in nearly all the techniques involved in Parkour. You will naturally have one foot you prefer to kick with on a wall run, one foot you find more comfortable to jump off for a speed vault, and one side you prefer to do lazy vaults on, this is normal, just as it is normal to prefer to write with a certain hand.
If you seek to obtain freedom of movement, the ability to adapt to your environment without thinking, and efficiency of movement, you can’t favour a side, arm, or leg. For example, if you favour a leg to take off from for a speed vault, it is likely that you will stutter step or make some kind of adjustment to your stride on approach to make sure that you favourite leg is in position jump from. This will cause you to slightly slow down moving into the vault. While this may seem like a minor thing to some people, it is inefficient and restrictive. Being able to approach that vault without breaking stride and take off from whichever foot comes down in position is the most natural and efficient way of overcoming the obstacle. It may not always possible to avoid some adjustments even if you can used both sides of the body, but being ambidextrous certainly helps reduce the need to make adjustments.
Another example which is more obvious is when performing a tic tac. Most people will have a foot they prefer to kick off the wall with, but there are situations where you have to do a tic tac and it will be on your weak side. Obviously, this can be a problem, and an obstacle in itself. If you want to reach a high level of skill, you will need to train out your weaknesses on certain sides.
It may not be the most fun to train something you can do comfortably on one side of the body, but struggle to do on the other, and sometimes it can be very frustrating, but it is a vital part of Parkour training. You will probably find there are certain techniques you can do reasonably well on both sides without a lot of attention, but there will be other techniques which will require a lot of work. You may have to slow it right down, walk into it, and really break down the elements involved in the technique just like learning how to do it for the first time. You may even improve your technique on your strong side, or gain a better understanding of the technique by having to re-learn it on your weak side.
There are no real secrets to training techniques on your weak side, if you’ve managed to learn it on one side, you should be able to do it on the other, a lot of it comes down to patience, commitment and focus. Often your weak side is actually physically weaker and can be the cause of favouritsm, so focusing on specifically strengthening your weak side can be helpful in becoming more well rounded in your ability to move. Different variations of quadrupedal movement can help with building overall, balanced strength and can also increase coordination between the limbs and the upper and lower body. Overall body coordination is very important in Parkour, particularly in being able to use both sides of the body equally well and not having favourites.
Ask yourself, can I roll over both shoulders, can I tick tack well with both legs, do I have to have a certain foot forward to do a cat pass, can I move freely without relying on a favourite side? If you answer no, go out, put your head down, and really focus on drilling something on your weak side. It may be frustrating, but the reward of being able to move without favouring a certain side is well worth it and your ability to move will greatly increase.