The Graded Dual of a Combinatorial Hopf Algebra on Partition Diagrams

John M. Campbell constructed a combinatorial Hopf algebra (CHA) $\text{ParSym}$ on partition diagrams by lifting the CHA structure of $\text{NSym}$ (the Hopf algebra of noncommutative symmetric functions) through an analogous approach. In this article, we define $\text{ParQSym}$, which is the graded dual of $\text{ParSym}$. Its CHA structure is defined in an explicit, combinatorial way, by analogy with that of the CHA $\text{QSym}$ of quasisymmetric functions. And we give some subcoalgebra and Hopf subalgebras of $\text{ParQSym}$, some gradings and filtrations of $\text{ParSym}$ and $\text{ParQSym}$, and some bases of $\text{ParSym}$ and $\text{ParQSym}$ by analogy with some distinguished bases of $\text{NSym}$ and $\text{QSym}$.