In Prasolov - Sossinsky there is an exercise about equivalent surgeries and there is this solution at the back of the book. But looks like there's a right Reidemeister 1 performed on the component with framing $r_2$ in the second shape. So shouldn't the framing change by a factor of $+1$?
Surgery coefficients are with respect to the $0$ framing of each component independently (that is, in a link $L=L_1\cup \dots\cup L_n$, the longitude for component $L_i$ is the curve in $\partial(S^3-\nu(L_i))$ that is nullhomologous in $S^3-L_i$), so doing a Reidemeister I move does not change the surgery coefficients, as this corresponds to an isotopy of the link.
(Reidemeister I changes the so-called "blackboard framing" by $\pm 1$. In the book, it looks like 15.6 explains the $0$ framing, the "invariant definition of the parallel on the tubular $\epsilon$-neighborhood of an arbitrary oriented knot $J$," and it does so in terms of linking numbers.)