I'm not sure Parliament is ready to abandon any pretext of "democracy" by completely ignoring this referendum.
The reality is that only 3.8% more people want to exit the EU than people that want to stay in the EU. When making decisions you have to factor in the opportunity costs, and with only a 3.8% preference for leaving I don't think it's worth the upheaval to change course. Something as major as this should require a supermajority vote. Here in the United States constitutional amendments require a 2/3 majority vote in both houses of Congress. Ratifying treaties requires a 2/3 majority vote of the Senate. Ratifying amendments requires a 3/4 majority vote of all the states. Scotland and Northern Ireland both want to stay in the EU. The final result could be within the margin of error for the statistical sample since only about 70% of the population voted. They should have another vote to decide if they should proceed with Article 50.