Optimal Transport (OT) has proven useful to infer single-cell trajectories of developing biological systems by aligning distributions across time points. Recently, Parameterized Monge Maps (PMM) were introduced to learn the optimal map between two distributions.Here, we apply PMM to model single-cell dynamics and show that PMM fails to account for asymmetric shifts in cell state distributions. To alleviate this limitation, we propose Unbalanced Parameterised Monge Maps (UPMM). We first describe the novel formulation and show on synthetic data how our method extends discrete unbalanced OT to the continuous domain. Then, we demonstrate that UPMM outperforms well-established trajectory inference methods on real-world developmental single-cell data.