We construct quasi-local conserved currents in the six-vertex model with anisotropy parameter η by making use of the quantum-group approach of Bernard and Felder. From these currents, we construct parafermionic operators with spin 1+iη/π that obey a discrete-integral condition around lattice plaquettes embedded into the complex plane. These operators are identified with primary fields in a c=1 compactified free Boson conformal field theory. We then consider a vertex-face correspondence that takes the six-vertex model to a trigonometric SOS model, and construct SOS operators that are the image of the six-vertex currents under this correspondence. We define corresponding SOS parafermionic operators with spins s=1 and s=1+2iη/π that obey discrete integral conditions around SOS plaquettes embedded into the complex plane. We consider in detail the cyclic-SOS case corresponding to the choice η=iπ(p−p′)/p, with p′<p coprime. We identify our SOS parafermionic operators in terms of the screening operators and primary fields of the associated c=1−6(p−p′)2/pp′ conformal field theory.