This is fundamentally a class distinction.
With any given amount of land and labor, more food value can be created from growing grain and vegetables than from growing animals for meat. In the medieval economy, the local lord had title to all the land and had a large amount of labor at his disposal as a sort of tax on his peasant subjects. The lord could thus afford to invest a portion of that land and labor into growing meat for his table. But the ordinary peasant family, with only their own labor and a small allotment of land, could not afford meat; they instead ate grain and vegetables.
For a couple of centuries after the Norman Conquest in 1066, the ruling class of England spoke mostly French, and spoke of the meat they ate using the French words for the animals: boeuf, porc, mouton. The peasants whose labor went into raising these animals for their lords' tables continued to call these animals by their native English names: cow, pig, sheep. And the lexical distinction remained after the landlord class adopted English. They had spoken of the food for several generations as beef, pork, mutton; and there was no corresponding term in English for the food, since the English words designated primarily the animal.
Neither is very common, except in speeches given by people learning English.
"There are four people in my family." is a simple and correct expression.
It is probably more common, if you are asked to talk about your family, to describe them:
Some families are simple
Tell me about your family
There's my wife and we have three children
You don't need to say "five people" because we can count.
On the other hand, some are more complex
Tell me about your family.
Well I live with my girlfriend and we have three kids, but the eldest is at university, Katy, my stepdaughter is at high school and the youngest, Jake, is still a baby. Also my girlfriend's mother lives in the annexe.
Families are complex: consider the example. How many people are in that family?
It would normally be "me" because that word is not the subject of a clause. In this respect, English is different from some other languages (in particular Latin and some Latin derived languages). There is some flexibility on this. "It is me." is normal and correct. "It is I" is also correct, but less used. In a long phrase "My mother, my father, my sister and I" could also be correct, but unnusual. English tends to reserve "I" for first-person singular subjects.
Best Answer
Damkerng T is right that "family" can be treated as singular or plural:
So:
or
These mean the same thing - it's just that, in 2), the speaker is emphasising the individual members of their family, rather than the family as a group. It really means "My family (members) are..."
Of course, your example sentence does not say "...that the family is ..."
Instead, it is giving two subjects: parents and family, which suggests that "are" is better here. Possibly the writer is thinking of parents as part of "the family", so groups both items together with the singular "is".
The fact remains, there are two subjects given, even if one is part of the other, so the plural "are" should follow (grammatically speaking).
To emphasise that one is part of the other, "the family (especially/including parents) is..." would be better.