I was going through the text "Discrete Mathematics and its Applications" by Kenneth Rosen where I came across the definition of bipartite graph, which assumes a the graph to be simple.
A simple graph $G$ is called bipartite if its vertex set $V$ can be partitioned into two disjoint sets $V_1$ and $V_2$ such that every edge in the graph connects a vertex in $V_1$ and a vertex in $V_2$ (so that no edge in G connects either two vertices in $V_1$ or two vertices in $V_2$). When this condition holds, we call the pair $(V_1,V_2)$ a bipartition of the vertex set $V$ of $G$.
Now from the condition that each edge should run only between two bipartitions so formed, we cannot have a self loop as it an edge from a vertex to itself in the same bipartition. But what restrictions do we have to include parallel edges running between the two bipartitions? I hope there is none. So we can have multigraphs as bipartite without violating the terms conveyed by the meaning of the definition.