Based on an open radio access network (O-RAN) architecture in terrestrial networks (TNs), this study proposed an open non-terrestrial network (open NTN) architecture in which multiple service operators could share low earth orbit (LEO) satellites for cost-effective network deployment in LEO-based NTNs. In the open NTN, a centralized unit (CU) of next-generation node B (gNB) is located in a gateway, and the distributed units (DUs) and radio units (RUs) are in LEO satellites. Inter-satellite links (ISLs) between LEOs function as wireless fronthaul links, where data and pilot symbols are delivered for further processing. Compared to O-RAN in TNs with wired fronthaul, the bandwidth constraints in wireless ISL fronthaul are more stringent, and signal compression in fronthaul cannot be avoided. Therefore, to improve the efficiency of open NTNs, a joint optimization problem of ISL fronthaul compression rate and power allocation to the data and pilot signals was formulated to maximize the sum of uplink rates of multiple user equipments (UEs) under the bandwidth constraints of the ISL fronthauls. The original problem was divided into two subproblems: one is an optimization of the data and pilot compression rate and the other is that of the data and pilot power allocation. An alternating optimization approach that solved two sub-problems in an iterative manner was adopted to find the convergent near-optimal solution. The convergence and performance of the proposed approach were verified with numerical results.