Set reconciliation is a fundamental algorithmic problem that arises in many networking, system, and database applications. In this problem, two large sets A and B of objects (bitcoins, files, records, etc.) are stored respectively at two different network-connected hosts, which we name Alice and Bob respectively. Alice and Bob communicate with each other to learn $AΔB$, the difference between A and B, and as a result the reconciled set $A\bigcup B$. Current set reconciliation schemes are based on either Invertible Bloom Filters (IBF) or Error-Correction Codes (ECC). The former has a low computational complexity of O(d), where d is the cardinality of $AΔB$, but has a high communication overhead that is several times larger than the theoretical minimum. The latter has a low communication overhead close to the theoretical minimum, but has a much higher computational complexity of $O(d^2)$. In this work, we propose Parity Bitmap Sketch (PBS), an ECC- based set reconciliation scheme that gets the better of both worlds: PBS has both a low computational complexity of O(d) just like IBF-based solutions and a low communication overhead of roughly twice the theoretical minimum. A separate contribution of this work is a novel rigorous analytical framework that can be used for the precise calculation of various performance metrics and for the near-optimal parameter tuning of PBS.