I guess the level needs to be designed so that the platforms don't overlap. I am not sure what you meant here
I meant that if the parent has to be at 1,1,1 then all the platforms should be at 1,1,1 being the parent. But then I finally got what you said very clearly, it's just me I didn't get it - what I need is a empty object at 1,1,1 which is child to the specific platform, but becomes parent to the object on the platform when it's on it. So I don't need the actual platform to be at 1,1,1.
And it worked. Thank you @Alex, @1982!
I wonder is the 1,1,1 a trick, a workaround, a bug or something deliberate by Unity.