Thirukkural is one of the famous Tamil Literatures in the world. It was written by Thiruvalluvar, and focuses on ethics and morality. It provides all possible solutions to lead a successful and a peaceful life fitting any generation. It has been translated into 82 global languages, which necessitate the access of Thirukkural in any language on the World Wide Web (WWW) and processing the Thirukkural computationally. This paper aims at constructing the Thirukkural Discourse Parser which finds the semantic relations in the Thirukkurals which can extract the hidden meaning in it and help in utilizing the same in various Natural Language Processing (NLP) applications, such as, Summary Generation Systems, Information Retrieval (IR) Systems and Question Answering (QA) Systems. Rhetorical Structure Theory (RST) is one of the discourse theories, which is used in NLP to find the coherence between texts. This paper finds the relation within the Thriukkurals and the discourse structure is created using the Thirukkural Discourse Parser. The resultant discourse structure of Thirukkural can be indexed and further be used by Summary Generation Systems, IR Systems and QA Systems. This facilitates the end user to access Thirukkural on WWW and get benefited. This Thirukkural Discourse Parser has been tested with all 1330 Thirukurals using precision and recall.