A graph $G$ has the \emph{strong parity property} if for every subset $X\subseteq V$ with $|X|$ even, $G$ has a spanning subgraph $F$ with minimum degree at least one such that $d_F(v)\equiv 1\pmod 2$ for all $v\in X$, $d_F(y)\equiv 0\pmod 2$ for all $y\in V(G)-X$. Bujtás, Jendrol and Tuza (On specific factors in graphs, \emph{Graphs and Combin.}, 36 (2020), 1391-1399.) introduced the concept and conjectured that every 2-edge-connected graph with minimum degree at least three has the strong parity property. In this paper, we give a characterization for graphs to have the strong parity property and construct a counterexample to disprove the conjecture proposed by Bujtás, Jendrol and Tuza.