What is the difference in usage between " and" and " or " in the parallel structure?

1. Students at school were told not to drink, not to smoke and not to fight.
2. Students at school were told not to drink, smoke or fight.

Are they ungrammatical? I'm quite confused about the usage of these two coordinating conjunctions, especially in the parallel structure.

I'll explain the difference in usage between "and" and "or" in such structures by using your two examples:

In example 1, "and" is used because each item in the list expresses the negative. Because "not to" is contained in each item of the list, using "and" means that students were told not to do all of those things.

Using "or" wouldn't be ungrammatical, but it would mean that students at the school weren't told all of those things not to do, so, for example, students after getting caught drinking and fighting could say that they did as they were told since they didn't smoke. That's because when told "not to drink, not to smoke, or not to fight," the use of "or" means they're only being told not to do one of those things, not not to do all of those things, and what that thing was or what they chose for that thing to be was "not to smoke," meaning it was therefore permissible for them to drink and fight.

In example 2, "or" is used because each item in the list doesn't express the negative but instead the negative prefaces the list.

Typically, "or" or "nor" instead of "and" is used in a list of items that are not themselves negatives but refer back to a negative that prefaces that list, a negative that is to be carried through that list, whereas if that list weren't prefaced with a negative, "and" would be used.

Using "and" in a list prefaced by a negative that the list refers back to is likewise not ungrammatical, but it does change the meaning since it can be perceived as meaning not to do all of those things instead of not to do any of those things, so someone doing just one or two of those things in your example could rightly claim that they did as they were told since they didn't "drink, smoke and fight" but, for example, only drank and smoked, the fact that they didn't fight making it true that they didn't "drink, smoke and fight."

Let A = to allow to drink, B = to allow to smoke, C = to allow to fight.

The first sentence is making the statement f1 = Ā•B̅•C̅

The second statement is f2 = (A+B+C)'

By DeMorgan's Law, these are logically equivalent (f1=f2). If you draw out the truth tables, they're identical.

To answer your other question, both are grammatically valid, too.