  In this paper, we provide exact formulas for the Castelnuovo-Mumford
regularity of powers of an almost complete intersection ideal $I$ which is
generated by a homogeneous $d$-sequence. As applications, when $I$ is an almost
complete intersection, taking the form of the (parity) binomial edge ideal of a
connected graph, we can describe explicitly formulas for $\reg(I^t)$ for $t\ge
2$. The only exception is when $I$ is the parity binomial edge ideal of a graph
which is obtained by adding an edge between two disjoint odd cycles.
