On a literal level the statement is of course bollocks.
My partner and I finally got around to getting married when our kids were at primary school. Whether we are good, bad or indifferent parents is perhaps for others to judge. One thing I feel confident in claiming though is that we definitely didn't suddenly get better on the day we tied the official knot.
It may be that couples who get married are more likely to be those who are already committed to a long term stable relationship to support their children -rather than it being marriage that creates that type of relationship i.e. the causal relationship is more the opposite to that which is being suggested,
On the other hand neighbourhoods where the norm is for couples to split up easily, with the almost inevitable consequence of many fatherless families, do seem to be unsatisfactory and turbulent ones, to an extent that can't be put down simply to poverty or deprivation. Neighbourhoods where there is a recognition that couples who have children are, or should be, in it for the long haul, do seem more stable, with better adjusted children - again to a greater extent than can be attributed just to relative wealth.
I don't know whether humans are unique in this respect in the animal kingdom. We are at least very unusual in that children take a decade or more of upbringing before they can look after themselves. You just can't kick them out of the nest and out onto the street on day 2 (not unless you want a social worker knocking on your door within 5 minutes). So bringing up a family is really a job for two people by and large. This is not to deny that there are many brilliant single parents - but the point is that they have to be to achieve the same level of stability.
I would say that couples who commit to each other and don't just split up as soon as things get a bit rocky do on the whole - not invariably of course - tend to be better parents. Marriage is an indicator rather than a cause of this.