Hi Erchenko Viktor,
Before I get to your question, I would like to comment that it sounds unnatural to say "The boyfriend of my sister".
We tend to use "the X (noun) of Y (noun)" for parts of inanimate objects (e.g. the timer of the machine, the gear of the truck, the settings of the computer). It would be more natural to say "My sister's boyfriend".
Secondly, there is a difference between "persuade" and "convince". Your sister's boyfriend is trying to make you believe and agree with his opinion, so he is trying to "convince" you. After someone manages to "convince" you, they can "persuade" you to do something.
Your sister's boyfriend is discussing the specific concept of "family" as a unit of society. Therefore, the specific article (i.e. the) should be used. As an analogy, someone can try to convince me that THE car will be irrelevant in the future when we are able to travel in flying pods. We are not referring to a specific car here; rather, we are referring to the specific mode of transport using cars.
"An" is used to specify "absolutely", not "institution". It is used when the pronunciation of the next word begins with a vowel sound (in the case of "absolutely", it starts with a vowel /æ/ sound). Sometimes, an adjective (or multiple adjectives) comes between the article and the noun. Despite this, the rule of using "an" when the next word is pronounced with a starting vowel sound still holds.
Other examples: An irritating boy --> "irritating" starts with a vowel /ɪ/ sound // An old-fashioned institution --> "old" starts with a vowel / oʊ/ sound.
I hope this helps you.