We resolve the explicit bijection problem between symmetric plane partitions (SPPs) and quasi transpose complementary plane partitions (QTCPPs), introduced by Schreier-Aigner, who proved their equinumerosity. First, we relate this problem to Proctor's parallel equinumerosities for SPPs, even SPPs, staircase plane partitions, and parity staircase plane partitions, by constructing several bijections. As a result, we reduce the task to constructing a compatible bijection between even SPPs and staircase plane partitions. We then provide non-intersecting lattice path configurations for these objects, apply the LGV lemma, and transform the resulting path configurations. This process leads us to new combinatorial objects, $I_m$ and $J_m$, and the task is further reduced to constructing a compatible sijection (signed bijection) between $I_m$ and $J_m$, which is carried out in the final part of this paper. Our construction also answers the 35-year-old open problem posed by Proctor: constructing an explicit bijection between even SPPs and staircase plane partitions.