Database Management System (DBMS) is the key component for data-intensive applications. Recently, researchers propose many tools to comprehensively test DBMS systems for finding various bugs. However, these tools only cover a small subset of diverse syntax elements defined in DBMS-specific SQL dialects, leaving a large number of features unexplored. In this paper, we propose ParserFuzz, a novel fuzzing framework that automatically extracts grammar rules from DBMSs' built-in syntax definition files for SQL query generation. Without any input corpus, ParserFuzz can generate diverse query statements to saturate the grammar features of the tested DBMSs, which grammar features could be missed by previous tools. Additionally, ParserFuzz utilizes code coverage as feedback to guide the query mutation, which combines different DBMS features extracted from the syntax rules to find more function and safety bugs. In our evaluation, ParserFuzz outperforms all state-of-the-art existing DBMS testing tools in terms of bug finding, grammar rule coverage and code coverage. ParserFuzz detects 81 previously unknown bugs in total across 5 popular DBMSs, where all bugs are confirmed and 34 have been fixed.