We introduce a method for unsupervised parsing that relies on bootstrapping classifiers to identify if a node dominates a specific span in a sentence. There are two types of classifiers, an inside classifier that acts on a span, and an outside classifier that acts on everything outside of a given span. Through self-training and co-training with the two classifiers, we show that the interplay between them helps improve the accuracy of both, and as a result, effectively parse. A seed bootstrapping technique prepares the data to train these classifiers. Our analyses further validate that such an approach in conjunction with weak supervision using prior branching knowledge of a known language (left/right-branching) and minimal heuristics injects strong inductive bias into the parser, achieving 63.1 F$_1$ on the English (PTB) test set. In addition, we show the effectiveness of our architecture by evaluating on treebanks for Chinese (CTB) and Japanese (KTB) and achieve new state-of-the-art results. Our code and pre-trained models are available at https://github.com/Nickil21/weakly-supervised-parsing.