I wonder if I can replace over with on in the sentence below and if the meaning remains the same. If there is any difference, I would like to ask for the explanation of it.
Her parents no longer have any real influence over her.
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The two expressions have little to distinguish them but influence over is more likely to be used in a parent-child or similar context and in a person to person situation.
You will come across statements such as: Her adolescent son is running wild and his mother no longer has any influence over him.
Influence on might be used in situations concerning a person, a group, an organisation or a decision.
Given that the two expressions overlap, it really comes down to context and preference.
Google Books Ngram Viewer indicates that influence on is several times more popular than influence over.