A second referendum doesn't just require one vote in Parliament though - it would require a Bill to be passed and then the government to manage it. Do you think its plausible that this government would do that?
Also, I would like to draw your attention to the fact that Labour under Corbyn did actually vote for a second referendum nearly (bar one SNP-led proposal) every time it came up. They also voted for the two amendments (Boles and Clarke's) that would have allowed Brexit but with either a customs union / SM membership that would have passed but for Lib Dem / TIG opposition.
Finally, Labour's position was not to face both ways. Labour's position was to point out that such a distinction is false, that many of the people who vote leave and who vote remain have the same problems, the same grievances and the same needs. Of course, the people who foster this division didn't like it.