  We explore possible superconducting states in $t_{2g}$ multi-orbital
correlated electron systems with strong spin-orbit coupling (SOC). In order to
study such systems in a controlled manner, we employ large-scale dynamical
mean-field theory (DMFT) simulations with the hybridization expansion
continuous-time Quantum Monte Carlo (CTQMC) impurity solver. To determine the
pairing symmetry, we go beyond the local DMFT formalism using parquet equations
to introduce the momentum dependence in the two-particle vertex and correlation
functions. In the strong SOC limit, a singlet, $d$-wave pairing state in the
electron-doped side of the phase diagram is observed at weak Hund's coupling,
which is triggered by antiferromagnetic fluctuations. When the Hund's coupling
is comparable to SOC, a two-fold degenerate, triplet $p$-wave pairing state
with relatively high $T_c$ emerges in the hole-doped side of the phase diagram,
which is associated with enhanced charge fluctuations. Experimental
implications to doped Sr$_{2}$IrO$_{4}$ are discussed.
