In the burgeoning realm of Internet of Things (IoT) applications on edge devices, data stream compression has become increasingly pertinent. The integration of added compression overhead and limited hardware resources on these devices calls for a nuanced software-hardware co-design. This paper introduces CStream, a pioneering framework crafted for parallelizing stream compression on multicore edge devices. CStream grapples with the distinct challenges of delivering a high compression ratio, high throughput, low latency, and low energy consumption. Notably, CStream distinguishes itself by accommodating an array of stream compression algorithms, a variety of hardware architectures and configurations, and an innovative set of parallelization strategies, some of which are proposed herein for the first time. Our evaluation showcases the efficacy of a thoughtful co-design involving a lossy compression algorithm, asymmetric multicore processors, and our novel, hardware-conscious parallelization strategies. This approach achieves a 2.8x compression ratio with only marginal information loss, 4.3x throughput, 65% latency reduction and 89% energy consumption reduction, compared to designs lacking such strategic integration.