Parallel corpora are ideal for extracting a multilingual named entity (MNE) resource, i.e., a dataset of names translated into multiple languages. Prior work on extracting MNE datasets from parallel corpora required resources such as large monolingual corpora or word aligners that are unavailable or perform poorly for underresourced languages. We present CLC-BN, a new method for creating an MNE resource, and apply it to the Parallel Bible Corpus, a corpus of more than 1000 languages. CLC-BN learns a neural transliteration model from parallel-corpus statistics, without requiring any other bilingual resources, word aligners, or seed data. Experimental results show that CLC-BN clearly outperforms prior work. We release an MNE resource for 1340 languages and demonstrate its effectiveness in two downstream tasks: knowledge graph augmentation and bilingual lexicon induction.