We give a polyomino characterisation of recurrent configurations of the sandpile model on the complete bipartite graph Km,n in which one designated vertex is the sink. We present a bijection from these recurrent configurations to decorated parallelogram polyominoes whose bounding box is a m×n rectangle. Other combinatorial structures appear in special cases of this correspondence: for example bicomposition matrices (a matrix analogue of set partitions), and (2+2)-free posets. A canonical toppling process for recurrent configurations gives rise to a path within the associated parallelogram polyominoes. We define a collection of polynomials that we call q,t-Narayana polynomials, the generating functions of the bistatistic (area,parabounce) on the set of parallelogram polyominoes, akin to Haglund's (area,hagbounce) bistatistic on Dyck paths. In doing so, we have extended a bistatistic of Egge et al. to the set of parallelogram polyominoes. This is one answer to their question concerning extensions to other combinatorial objects. We conjecture the q,t-Narayana polynomials to be symmetric and discuss the proofs for numerous special cases. We also show a relationship between the q,t-Catalan polynomials and our bistatistic (area,parabounce)on a subset of parallelogram polyominoes.