How to draw the "parallel circuits" sign "//"?

You want that the double bar behaves like a binary operation, as far as spacing is concerned. So

\newcommand{\parallelsum}{\mathbin{\|}}

if you like vertical bars, or

\newcommand{\parallelsum}{\mathbin{\!/\mkern-5mu/\!}}

if you prefer slanted bars.

Wikipedia uses \| but doesn't distinguish it from \parallel. Personally, I'd use \| in text and \parallel in display math, since the latter has operator spacing.

IMHO \parallel perfectly suits the given situation. Alternatively, you may try \shortparallel from the amssymb package or \parallelslant from fourier. From my experience, when two circuit elements are connected in parallel, the formula for total value is written 'R = (R1^-1 + R2^-1 )^-1', as you put it, or 'R = R1 x R2' (using e.g. \times), the notation with \parallel is less popular with engineers.