but can you specify how can that explanation make sense if 2 charges are moving in parallel direction, both at rest in each other's frame?
Case 1: The distance between the charges is the same in the frame of the charges and the frame of the observe that observes the charges moving:
Special relativity tells us that we must divide the force felt by the charges by gamma, in order to get that same force in another frame. The force between charges in frame A is different than the force between those charges in frame B. When observers in frames A and B say "the force between the charges", they refer to the same force, which is different in those two frames.
Calculating the force felt by the charges is trivial in this case. Of course we need special relativity to know whether the distance is the same in both frames or not.
Case 2: distance between charges is not the same in both frames:
Special relativity tells us we must divide the force felt by charges by gamma, to get that same force in another frame. And special relativity tells us how to calculate the force felt by the charges. The force felt by charges is inversely proportional to distance squared. And Special relativity tells us what the distance is in the frame of the charges, then we can easily calculate the force felt by the charges.
In my opinion all that above stuff is trivial, so it makes a lot of sense.