When asking this type of question, the assumption is that you weren't doing any of these things before.
So, when you ask if something made you cry, the assumption is that you weren't crying before the thing happened.
But in this case, it's seems odd to assume that something made somebody silent. The "default" behaviour of somebody is to already be silent, a state from which they are made to do something else. This is certainly not always the case, but it's the more common of situations.
Given that, I am going to take the liberty of rephrasing your question slightly—so that, a different context aside, neither version sounds odd when it comes to silence.
Are both of the following sentences correct?
Did it make you laugh or make you remain silent?
Did it make you laugh or remain silent?
Both of these sentence are grammatical. I prefer the first to the second, but the second isn't wrong.
I should add that I have a third phrasing. In this particular case, because of the "silent" part, I prefer it to the other two:
Did it make you laugh or keep you silent?
To directly answer your original question, if your first example sentence is acceptable, then so is your second example sentence. But I simply find them too awkward to leave as they are.
I acknowledge the fact that there is a problem with parallelism, however I find that this just adds to the awkwardness rather than producing something that is actually incorrect syntactically. It's not something that, purely on its own, would be noticeable by most people—and the other issue with awkwardness outweighs it.
But if the need for parallelism is something that's a requirement, then neither of the example sentences are correct.
The second sentence is simply an elliptical version of the first:
Did it make you laugh or make you silent?
Did it make you laugh or [make you] silent?
The second make you is assumed to be there, even though it has been omitted. (In reality, the first sentence also has an assumed [did it] after the or.)
This is the same as (in reverse):
I have washed and ironed my clothes.
I have washed [my clothes,] and [I have] ironed my clothes.)
The conjunction joins one clause with an assumed second clause.
The only time when such this wouldn't happen would be if the conjunction actually signals a singular collective:
I ate the fish and chips.
There are two ways of parsing this:
I ate the fish and chips.
I ate the fish [and I ate the] chips.
In this case, it's fish and chips that's correct because it's a collective singular term, and it's not the case that fish and chips are two separate things. (They are in one sense, but not in the sense we commonly mean it.)
Laugh or silent is not the same as fish and chips. They are two separate things that have been temporarily joined together for the sake of simplicity. In other words, it is an example of an elliptical sentence.
So, either the first and second sentences are both correct, or they are both wrong—because the second sentence means the same thing as the first.