Do formulas involving fewer repetitions of variables give higher numerical precision?
First, I want to say that it is not the case in general that an algorithm that minimizes the number of uses of the inputs is more accurate, at least for IEEE 754 floating point.  For example, compensated summation.  

On the other hand, it's certainly the case that interval arithmetic can greatly benefit from knowing when two inputs are identical.  As a trivial example, if $X$ is an interval variable, then logically $X - X = 0$ but of course the subtraction algorithm (usually) doesn't know whether it's inputs are logically identical, and so it must assume the worst-case $$X - X = [X^{lo} - X^{hi},X^{hi} - X^{lo}]$$ 

This is roughly what's going on here and almost certainly view SICP is taking.  If you use naive interval arithmetic operations, repeated uses of a variable will be treated independently, and so it won't be possible to cancel out error and the worst-case must be assumed.  Using the formulas from the interval arithmetic Wikipedia page and assuming $R_1$ and $R_2$ are represented by intervals that are strictly positive, you can simply calculate:
$$\frac{1}{\frac{1}{R_1}+\frac{1}{R_2}} = \left[\frac{1}{\frac{1}{R_1^{lo}}+\frac{1}{R_2^{lo}}},\frac{1}{\frac{1}{R_1^{hi}}+\frac{1}{R_2^{hi}}}\right]$$
while
$$\frac{R_1 R_2}{R_1 + R_2} = \left[\frac{R_1^{lo}R_2^{lo}}{R_1^{hi}+R_2^{hi}},\frac{R_1^{hi}R_2^{hi}}{R_1^{lo}+R_2^{lo}}\right]$$

It's clear that $$\frac{R_1^{lo}R_2^{lo}}{R_1^{hi}+R_2^{hi}} \leq \frac{R_1^{lo}R_2^{lo}}{R_1^{lo}+R_2^{lo}}$$ and symmetrically for the upper bounds, so the latter formula has a looser interval.