Genome-wide association studies of neurological diseases have identified thousands of variants associated with disease phenotypes. However, most of these variants do not alter coding sequences, making it difficult to assign their function. Here, we present a multi-omic epigenetic atlas of the adult human brain through profiling of single-cell chromatin accessibility landscapes and three-dimensional chromatin interactions of diverse adult brain regions across a cohort of cognitively healthy individuals. We developed a machine-learning classifier to integrate this multi-omic framework and predict dozens of functional SNPs for Alzheimer's and Parkinson's diseases, nominating target genes and cell types for previously orphaned loci from genome-wide association studies. Moreover, we dissected the complex inverted haplotype of the MAPT (encoding tau) Parkinson's disease risk locus, identifying putative ectopic regulatory interactions in neurons that may mediate this disease association. This work expands understanding of inherited variation and provides a roadmap for the epigenomic dissection of causal regulatory variation in disease.