''the pride of sth'' means a person or a thing that gives people a feeling of pleasure and satisfaction. So in that sense,as a young graduate,''the pride of her parents' and kindreds'/relatives' '' would be correct I guess.
Parents is a plural word formed by adding s.
Possessive form, parents'.
Kindred is a collective noun.
Possessive form, kindred's.
So you have a choice, assuming that you are referring to both parents
Using the possessive forms, you write:
... her parents' and kindred's pride. <- with apostrophes
Otherwise you write:
... the pride of her parents and kindred. <- without apostrophes