Building computational resources and tools for the under-resourced languages is strenuous for any Natural Language Processing task. This article presents the first dependency parser for an under-resourced Indian language, Nepali. A prerequisite for developing a parser for a language is a corpus annotated with the desired linguistic representations known as a treebank. With an aim of cross-lingual learning and typological research, we use a Bengali treebank to build a Bengali-Nepali parallel corpus and apply the method of annotation projection from the Bengali treebank to build a treebank for Nepali. With the developed treebank, MaltParser (with all algorithms for projective dependency structures) and a Neural network-based parser have been used to build Nepali parser models. The Neural network-based parser produced state-of-the-art results with 81.2 Unlabeled Attachment Score, 73.2 Label Accuracy, and 66.1 Labeled Attachment Score on the gold test data. The parser models have also been evaluated with the predicted Part-of-speech (POS)-tagged test data. A statistical POS tagger using Conditional Random Field has been developed for predicting the POS tags of the test data.