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.
At least for the GitHub Flavoured Markdown, you can give the illusion by making all the non‑header row entries bold with the regular __ or ** formatting: