Certainly a strange request by your teachers. I can only think of defining your own \cite command to use when you're inside a parenthesis.
I assume you would require the square versions less, so I pass a default round option to natbib, then define respective commands: \sqcitet and \sqcitep which are just wrappers to change (locally) the cite style.
\begin{filecontents}{\jobname.bib}
@ARTICLE{feyn54,
title = {Testing},
author = {R. P. Feynman},
journal = {Phys. Rev.},
year = {1954},
volume = {94},
pages = {262}
}
\end{filecontents}
\documentclass{article}
\usepackage[round]{natbib}
\makeatletter
\newcommand{\bianca}{\renewcommand\NAT@open{[}\renewcommand\NAT@close{]}}
\makeatother
\newcommand*\sqcitet[1]{{\bianca\citet{#1}}}
\newcommand*\sqcitep[1]{{\bianca\citep{#1}}}
\begin{document}
We would like to cite some in-line \citet{feyn54} and \citep{feyn54}, and of course in brackets as well:
(with \sqcitet{feyn54} and \sqcitep{feyn54})
We would like to cite some in-line \citet{feyn54} and \citep{feyn54}
\bibliographystyle{plainnat}
\bibliography{\jobname}
\end{document}

Extra Note
The documentation offers two ways to change the citation style, viz. (i) \setcitestyle{square} and (ii) creating a new \bibstyle@xxx and calling it with \citestyle{xxx}.
Method (i) is simple enough to implement, while implementation of Method (ii) can be seen in my previous edit of this answer. But..
The first method adds spurious spacings, while the second method sets the style globally. (and so both are unsatisfactory).
And so instead, I've gone for the option of explicitly setting the open and close parenthesis with \renewcommand\NAT@open{[} and \renewcommand\NAT@close{]} for the time being.
(Rankin, 2003 and Ringe, 2013)or(Rankin 2003 and Ringe 2013). Anyway, I don't think thatnatbiboffers an option to use both round parentheses and square brackets. I don't know aboutbiblatex, though. (Almost anything can be (re)configured withbiblatex...) – Mico Aug 09 '17 at 10:35natbibthere most likely is a way to switch this in the document, too, so one could build some custom macros with it. But I don't knownatbibthat well. Automated isn't reliably possible, I guess. – Skillmon Aug 09 '17 at 10:48