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We introduce area, bounce and dinv statistics on decorated parallelogrampolyominoes, and prove that some of their q,t-enumerators match $\langle\Delta_{h_m} e_{n+1},s_{k+1,1^{n-k}}\rangle$, extending in this way the work in(Aval et al. 2014). Also, we provide a bijective connection between decoratedparallelogram polyominoes and decorated labelled Dyck paths, which allows us toprove the combinatorial interpretation of the coefficient $\langle\Delta_{e_{m+n-k-1}}'e_{m+n},h_m h_n\rangle$ predicted by the Delta conjecturein (Haglund et al. 2015). Finally, we define a statistic pmaj on partiallylabelled Dyck paths, which provides another conjectural combinatorialinterpretation of $\Delta_{h_{\ell}}\Delta_{e_{n-k-1}}'e_n$, cf. (Haglund etal. 2015).  This is the full version of (D'Adderio, Iraci 2017) arXiv:1711.03923.

Parallelogram polyominoes, partially labelled Dyck paths, and the Delta conjecture (FULL VERSION)