Order of precedence for parallel operator
It looks like the parallel operator is surprisingly common, and there are precedence rules defined by some sources, but not all.

MIT in their courseware suggest using it, and set exam questions on it. They provide no information regarding precedence in their lecture notes. They say the following, with no description of precedence.


  This mathematical relationship comes up often enough that it actually has a name: the “parallel operator“,denoted by the symbol \$\parallel\$.  When we say \$x‖y\$, it means \$\frac{xy}{x+y}\$


The university of Utah use it to analyse circuits in their course notes, and provide precedence information as follows:


  In    the    absence    of    parentheses,    the    parallel    operators    act    before    the    plus    signs,    just    as    multiplies    act    before    the    plus    signs    in    a    calculator expression. In    the    parlance    of    computer    science,    "the    parallel    operator    binds    more    tightly    than    the    plus    operator."


Finally, this book also defines the operator, and provides precedence rules, in the same way when referring to the equation 
$$I = \frac{V_{in}}{R_1+R_2 \parallel R_3},$$ 


  the parallel operator has precedence over the addition: \$R_2 \parallel R_3\$ is computed first and then added to \$R_1\$


Note that neither of the references that define precedence of the \$\parallel\$ operator with respect to addition do so with respect to multiplication.

As I have rarely come across this operator, I would make no assumptions regarding precedence, and use brackets to clear up any ambiguity.
E.g.: $$I = \frac{V_{in}}{R_1+(R_2 \parallel R_3)}$$