A convex parameterization of internally stabilizing controllers is fundamental for many controller synthesis procedures. The celebrated Youla parameterization relies on a doubly coprime factorization of the system, while the recent system-level and input-output parametrizations require no doubly coprime factorization, but a set of equality constraints for achievable closed-loop responses. In this article, we present explicit affine mappings among Youla, system-level, and input-output parameterizations. Two direct implications of these affine mappings are: 1) any convex problem in the Youla, system-level, or input-output parameters can be equivalently and convexly formulated in any other one of these frameworks, including the convex system-level synthesis; 2) the condition of quadratic invariance is sufficient and necessary for the classical distributed control problem to admit an equivalent convex reformulation in terms of either Youla, system-level, or input-output parameters.