Here's an update on my research and how it went in my tank.
I searched many sources for the cause of those algae and how to kill them. It was suggested to me that red algae appear in high pH water, which wasn't my case (around 7). I haven't found a cause for them to appear.
Here's what I did to remove them: I had quite a bit of brown/green algae on the glass on my tank and I also had many plants hiding a great quantity of snails. Some of those plants weren't as healthy as they could be. So I cleaned my glass as best I could of algae. After that I removed all plants which were not super healthy (this step also removed several snails). I made a bigger water change right after (approx 40% vs my normal 25%). Less than a week after, I made a normal water change (~25%). The red algae regressed by themselves by the next week and as I stand now I don't have any left.
If they wouldn't have regressed I would have used this method I found here on a forum in French (in case you want to read/translate by yourself). The method is the peroxide one. If you resort to this method be sure you are ready to lose some/all your plants. Using a 3,5% peroxide you make a solution 1/9 with tap water (9 parts of water for each part of peroxide) and dip the plant in the solution. The time to leave the plant in the solution varies depending on the plant type. Plants with small leaves are to stay less than 1 minute, plants with normal leaves less than 2 minutes and plants with thick leaves less than 3 minutes. If you are not sure about the plant type, better to underdo it since too long in the solution will kill the plant. Right after the dip, you must rinse the plant as much as you can before you put it back in your tank. The algae will not appear dead right away but over the next couple of days it should start to discolor. As soon as the algae are dead, snails/shrimps/fish will start to eat it (which they wouldn't do normally when it's alive).