But push-down automata are significant not only because they have practical uses in parsing, but because they represent a theoretical class. They are more powerful than finite automata, but are not Turing complete.
If, say, we use a push-down automaton to make a Boolean decision: does this input string fall into the to-be-recognized set, or not? then there are some kinds of strings it won't be able to decide, that a full Turing machine could decide.
The limitation of push down automata is a direct consequence of the stack discipline. There is only one stack, and when the stack is popped, information is thrown away.
The tape machine that Turing described as part of developing his theory of computation doesn't have that limitation; when the tape head moves backwards, material it has written to the tape is not erased in a stack-like discipline.