The Encyclopedia of DNA Elements (ENCODE) and the Roadmap Epigenomics Project seek to characterize the epigenome in diverse cell types using assays that identify, for example, genomic regions with modified histones or accessible chromatin. These efforts have produced thousands of datasets but cannot possibly measure each epigenomic factor in all cell types. To address this, we present a method, PaRallel Epigenomics Data Imputation with Cloud-based Tensor Decomposition (PREDICTD), to computationally impute missing experiments. PREDICTD leverages an elegant model called "tensor decomposition" to impute many experiments simultaneously. Compared with the current state-of-the-art method, ChromImpute, PREDICTD produces lower overall mean squared error, and combining the two methods yields further improvement. We show that PREDICTD data captures enhancer activity at noncoding human accelerated regions. PREDICTD provides reference imputed data and open-source software for investigating new cell types, and demonstrates the utility of tensor decomposition and cloud computing, both promising technologies for bioinformatics.