The answers to this question got me thinking. Aunt Petunia probably spent her childhood with Lily and though she was bitter, was probably in close contact with her at least until she finished Hogwarts. Wouldn't she have known beforehand that they weren't allowed to use magic outside of school?
Lily actually broke the rules several times and did magic outside of school. As Petunia screams in Philosopher's Stone:
JK Rowling was apparently asked about this enough that it became an FAQ on her website, in which she reveals that Aunt Petunia was telling the truth. Lily really did do magic outside of school.
Petunia may have seen the magic as a child, but not realized what the letters Lily was receiving were saying.
She definitely knew once, but seems to have forgotten.
We know from memories we see in the final book that Petunia overheard a conversation in which Lily was told about the Statute of Secrecy, and how she couldn’t do magic outside school:
So she knew about this rule at some point.
But when Hagrid arrives to take Harry to school (about twenty years later), the way Petunia describes it sounds as if Lily was doing magic outside school:
When Harry receives his letter from the Ministry, it also sounds as if all the Dursleys, including Petunia, are concerned that Harry would use magic outside school:
So it seems that she really has forgotten.
I would guess that when she married Vernon, Petunia tried to push all knowledge of the magical world to the back of her mind. It may also have helped if Vernon, paranoid as he is, kept talking about the possibility that Harry would use his magic against them.
Alternatively, given that Vernon and Petunia don’t set much store by Harry, his parents or the magical world in general, she could have convinced themselves that Harry would ignore this law without any real consequence, so legitimising the fear.
I would also like to point out that even though Harry isn't allowed to do magic outside of school, he is still capable of it. It's not like they had any way of stopping him if he tried.
But I agree more with the above answers in that she merely forgot. It never pertained to her and she did see Lily doing magic.
Side note: I would wager that small magic of no consequence would have been overlooked/ not even reported to the ministry. Hermione, when she first met the boys on the train says something along the lines of
(if anyone has the direct quote feel free to edit this, I don't have the books with me.) Indicating that even with a wand/ being 11 and of Hogwarts age, it's not like kids weren't casting spells. But really, who wants to expell a kid for turning tea cups into rats (and probably bad ones with them being so young)
Of course Petunia would have no way of classifying spells and as far as she was concerned, even if she remembered the rules, they weren't enforced.