Semigroups of order-preserving transformations have been extensively studied for finite chains. We study the monoid $OP_{\mathbb{N}}$ of all order-preserving partial transformations on the set $\mathbb{N}$ of natural numbers, where the partial order is a fence (also called zigzag poset). The monoid $OP_{\mathbb{N}}$ is not regular. In this paper, we determine particular maximal regular subsemigroups of $OP_{\mathbb{N}}$ and show that $OP_{\mathbb{N}}$ has infinitely many maximal regular subsemigroups.