It depends what you mean by "father of linguistics".
Pāṇini's Aṣṭādhyāyī is the oldest surviving work that could be called a complete linguistic grammar. But Pāṇini wasn't writing in a vacuum: his work references earlier works that haven't survived, by other authors. It's mostly a historical accident that his grammar survived when none of his predecessors' did.
However, Ferdinand de Saussure is often called one of the fathers of linguistics (i.e. one of the people who turned European/Western linguistics from a pastime into a science). Sanskrit was one of his specialties, and he specifically cited Pāṇini's work as an influence on his own.
Noam Chomsky, who's also sometimes called the father of linguistics, was then deeply influenced by Saussure—many of Chomsky's more influential ideas came about from either defending or refuting Saussure's theories.
And nowadays, it's hard to find any area of linguistics that hasn't been affected (for better or for worse) by Chomsky's work. So on the one hand, yes, pretty much all of modern linguistics has been influenced indirectly by Pāṇini, and some ideas introduced(?) by Pāṇini are now so standard we don't even think about them (such as formal systems and auxiliary symbols). On the other hand, we don't know how much of that is due to Pāṇini and how much is due to his predecessors whose complete works haven't survived, and linguistics as a science in and of itself didn't really exist until the late nineteenth century, long after his death.
So, how much of that you want to attribute to Pāṇini himself is up to you.