Can deep learning be applied to nonlinear parameter estimation problems?
No.  Deep learning is not useful or relevant here.  Deep learning is for classification.  You don't have a classification task.  Rather, you have an optimization task: find the parameter value $p$ that causes the best match between the given $x$-values and $y$-values.

Deep learning isn't magic sauce that you can apply to anything you might ever think of and expect it to solve your problem.  Your question is an instance of the XY problem.  There may be suitable methods to speed up optimization, but you'll probably need to ask a new question and tell us something about the structure/properties of $f$ and the typical range of values for $n_i$.