Histone deacetylases (HDACs) have been widely pursued as targets for anti-cancer therapeutics. However, many of these targets are universally essential for cell survival, which may limit the therapeutic window that can be achieved by drug candidates. By examining large collections of CRISPR/Cas9-based essentiality screens, we discovered a genetic interaction between HDAC1 and HDAC2 wherein each paralog is synthetically lethal with hemizygous deletion of the other. This collateral synthetic lethality is caused by recurrent chromosomal translocations that occur in diverse solid and hematological malignancies, including neuroblastoma and multiple myeloma. Using genetic deletion or dTAG-mediated degradation, we show that HDAC2 disruption suppresses the growth of HDAC1-deficient neuroblastoma in vitro and in vivo. Mechanistically, we find that targeted degradation of HDAC2 in these cells prompts the degradation of several members of the nucleosome remodeling and deacetylase (NuRD) complex, leading to diminished chromatin accessibility at HDAC2/NuRD-bound sites of the genome and impaired control of enhancer-associated transcription. Furthermore, we reveal that several of the degraded NuRD complex subunits are dependencies in neuroblastoma and multiple myeloma, providing motivation to develop paralog-selective HDAC1 or HDAC2 degraders. Altogether, we identify HDAC1/2 collateral synthetic lethality as a new therapeutic target and reveal a novel mechanism for exploiting NuRD-associated cancer dependencies.