Let R be a semiprime ring with extended centroid C and let
$I(x)$
denote the set of all inner inverses of a regular element x in R. Given two regular elements
$a, b$
in R, we characterise the existence of some
$c\in R$
such that
$I(a)+I(b)=I(c)$
. Precisely, if
$a, b, a+b$
are regular elements of R and a and b are parallel summable with the parallel sum
${\cal P}(a, b)$
, then
$I(a)+I(b)=I({\cal P}(a, b))$
. Conversely, if
$I(a)+I(b)=I(c)$
for some
$c\in R$
, then
$\mathrm {E}[c]a(a+b)^{-}b$
is invariant for all
$(a+b)^{-}\in I(a+b)$
, where
$\mathrm {E}[c]$
is the smallest idempotent in C satisfying
$c=\mathrm {E}[c]c$
. This extends earlier work of Mitra and Odell for matrix rings over a field and Hartwig for prime regular rings with unity and some recent results proved by Alahmadi et al. [‘Invariance and parallel sums’, Bull. Math. Sci.10(1) (2020), 2050001, 8 pages] concerning the parallel summability of unital prime rings and abelian regular rings.