The concept of concrete regularity structure gives the algebraic backbone of the operations involved in the local expansions used in the regularity structure approach to singular stochastic partial differential equations. The spaces and the details of the structures depend on each equation. We introduce here a parameter-dependent universal algebraic regularity structure that can host all the regularity structures used in the study of singular stochastic partial differential equations. This is done by using the correspondence between the notions of model on a regularity structure and the notion of paracontrolled system. We prove that the iterated paraproducts that form the fundamental bricks of paracontrolled systems have some local expansion properties that are governed by this universal structure.