As I can suppose by Rimmer's comment, the difference is visible only to mathematicians. We could move parts of the both sentences more apart.
It is possible, that the sentence "one can be happy only if one can be free" is true. I am not sure.
It is possible that one can be happy. But that is so if only this very person can be free. I.e. It is possible, if he is free.
But so it is well seen, that these both sentences are very close. One is probably true if the other is true and vice versa. In the normal language it is enough to feel them equivalent. (of course, they are not, but that doesn't matter. People don't use mathematical logics usually)
If you need to check how some constructions differ, you should find an example (better more than one), in which these two constructions have different meanings. If it is possible, they are different.
You are asking for even more elaborated meta-thinking. You see here two readings/constructions in the same sentence. The receipt is the same - find an example where they will really differ and then people will help you.
Let's try to change some words:
It is natural that one can be happy only if he is free.
For this sentence, according to my feelings, only the first reading is possible.
One can be happy if he is free. It is natural.
And not at any rate:
It is natural to be happy if only one is free. (If he is not free, his happiness is disgustful.)
So as I see, the first "parsing" would be more natural. Maybe, it could be changed by commas and that is the case where they are really very significant.