Given a graph $G$, we consider a model for a random cover of $G$ by taking two parallel copies of $G$ and crossing every pair of parallel edges randomly with probability $q$ independently of each other. The resulting graph $G_q$, is a random $2$-lift of $G$ that may not be transitive but still probabilistically exhibit many properties of transitive graphs. Studying percolation in this context can help us test the reliability and robustness of our proofs methods in percolation theory. Our three main results on this model are the continuity of the critical parameter $p_c(G_q)$, for $q\in(0,1)$, the strict monotonicity $p_c(G_q)< p_c(G)$ and the exponential decay of the cluster size in the subcritical regime at $q=1/2$.