To ensure coherent signal processing across distributed Access Points (APs) in Cell-Free Massive Multiple-Input Multiple-Output (CF-mMIMO) systems, a fronthaul connection between the APs and a Central Processor (CP) is imperative. We consider a fronthaul network employing parallel radio stripes. In this system, APs are grouped into multiple segments where APs within each segment are sequentially connected through a radio stripe. This fronthaul topology strikes a balance between standard star and bus topologies, which deploy parallel or serial connections of all APs. Our focus lies in designing the uplink signal processing for a CF-mMIMO system with parallel radio stripes. We tackle the challenge of finite-capacity fronthaul links by addressing the design of In-Network Processing (INP) strategies at APs. These strategies involve linearly combining received signals and compressing the combining output for fronthaul transmission, aiming to maximize the sum-rate performance. Given the high complexity and the stringent requirement for global Channel State Information (CSI) in jointly optimizing INP strategies across all APs, we propose an efficient sequential design approach. Numerical results demonstrate that the proposed sequential INP design achieves a sum-rate gain of up to 82.92% compared to baseline schemes.