Short answer: We are the American people, and because he is a federal legislator, Weiner works for us.
Seriously, has the left sunk so low that it now considers only illegality to be a meaningful bar to continued public service? If so, it's no wonder so many liberals want a nanny state micromanaging our every move. If one can't count on the good judgment or morals of one's fellow citizens to regulate their behavior, then there's no alternative to an intrusive, overbearing government to control one's every action.
But consider this: Had some US representative sent texts to women (some completely unknown to him) containing, say, egregious, disgusting anti-black or anti-woman remarks (perhaps, for argument's sake, despicable racist comments or pictures about Michelle Obama), chances are that Pareene -- and most of the left -- would call for his resignation. And rightly so. Such behavior, though not illegal, is unworthy of the House of Representatives, bringing shame on the House and the nation it governs.
So ultimately the question of resignation, absent illegality, "just" becomes a question of one's morals. Apparently, Pareene (and some of the other liberal writers whose views are discussed in the piece) simply don't think that texting pictures of one's privates to random women (probably some unsolicited and perhaps some underage), then lying about it and scurrilously blaming it on a political adversary is sufficiently shameful and dishonorable to warrant Weiner's resignation from the US House of Representatives.