Your code should already work, but it might be more reliable to access the property className instead:
jQuery(function($) { console.log($('.nav a.current').parent().prop('className'));});<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.2.4/jquery.min.js"></script><div class="nav"><ul><li class="tab1"><a href="#1" class="current">Welcome</a></li><li class="tab2"><a href="#2">About</a></li><!-- and some lists more --></ul></div>