Most Markdown parsers don't support tables without headers. That means the separation line for headers is mandatory.
Parsers that do not support tables without headers
multimarkdown
Maruku: A popular implementation in Ruby
byword: "All tables must begin with one or more rows of headers"
PHP Markdown Extra "second line contains a mandatory separator line between the headers and the content"
RDiscount Uses PHP Markdown Extra syntax.
GitHub Flavoured Markdown
Parsedown: A parser in PHP (used e.g. in Laravel emails)
Parsers that do support tables without headers.
Kramdown: A parser in Ruby
Text::MultiMarkdown: Perl CPAN module.
MultiMarkdown: Windows application.
ParseDown Extra: A parser in PHP.
Pandoc: A document converter for the command line written in Haskell (supports header-less tables via its simple_tables and multiline_tables extensions)
Flexmark: A parser in Java.
CSS solution
If you're able to change the CSS of the HTML output you can however leverage the :empty pseudo class to hide an empty header and make it look like there is no header at all.