Legitimacy is not a scientific concept; it is a normative concept. As a saying goes, legitimacy is in the eye of the beholder. In order words, legitimacy depends on an individual's perspective.
Let's look at our system's history lesson, which is something more solid. In Canadian history, a number of electoral reforms have been carried out. They began in 1920 in Manitoba and ended in 1956 in Alberta.
I have looked at the circumstances in which every one of those reforms was adopted. In each case—so in Alberta, in Manitoba and in British Columbia—the provincial Parliament implemented a reform without a referendum. At that time, holding a referendum was not even considered. Based on the customs of the time, it seems fine that it happened this way.
Those are the indications I can give regarding whether Parliament currently has the democratic legitimacy to proceed. Ours is a system of representative democracy. There is no legal obligation to hold a referendum, but it may occasionally happen that what can be done legally is perceived as illegitimate by a good portion of the population.
When it comes to that, I would like to emphasize that I am deeply troubled by the fact that, according to the four surveys I have looked at, many Canadians feel, rightly or wrongly, that a referendum should be held.