The coexistence of various low-lying deformed states in $^{42}$Ca and $alpha$--$^{38}$Ar correlations in those deformed states have been investigated using deformed-basis antisymmetrized molecular dynamics. Wave functions of the low-lying states are obtained via parity and angular momentum projections and the generator coordinate method (GCM). Basis wave functions of the GCM calculation are obtained via energy variations with constraints on the quadrupole deformation parameter $beta$ and the distance between $alpha$ and $^{38}$Ar clusters. The rotational band built on the $J^pi = 0_2^+$ (1.84 MeV) state as well as the $J^pi = 0_3^+$ (3.30 MeV) state are both reproduced. The coexistence of two additional $K^pi = 0^+$ rotational bands is predicted; one band is shown to be built on the $J^pi = 0_3^+$ state. Members of the ground-state band and the rotational band built on the $J^pi = 0_3^+$ state contain $alpha$--$^{38}$Ar cluster structure components.
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