Using the powerful machinery available for reduced words of type $B$, we demonstrate a bijection between centrally symmetric $k$-triangulations of a $2(n + k)$-gon and plane partitions of height at most $k$ in a square of size $n$. This bijection can be viewed as the type $B$ analogue of a bijection for $k$-triangulations due to L. Serrano and C. Stump.