In my cosmology textbook the author sometimes refers to the scale factor as $R\left(t\right)$ and sometimes just as plain $R$. For example:$$ds^{2}=c^{2}dt^{2}-R^{2}\left(t\right)\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta d\phi^{2}\right],$$$$H\left(t\right)=\frac{1}{R}\frac{dR}{dt}.$$
I can't see any any rhyme or reason to his usage. Now, it's not important, I know what he means, but is there a rule or convention governing when you should or shouldn't use parentheses in function notation?
No there isn't. It is all about readability and, perhaps, reminding yourself occasionally that R depends on t. In the second equation such a reminder is superfluous as a derivative is being taken.