If I understand correctly, what confuses you is the use of past perfect (had gotten/had received) instead of past simple (got/received).
The typical use of past perfect is a past action completed before another past action. You receiving the email was before her making sure of it. That is why if making sure of it is in the past, receiving it should be even earlier in the past, which means past perfect.
So yes, her sentence is grammatically the best choice here. Yours would be a very commonly used one, which does not follow the correct grammar rules. She could have also said "I just want to be sure that you have got it." (Which is a version of what she actually used, only avoiding the past tense.)
Damkerng T is right that "family" can be treated as singular or plural:
Collective nouns, or collectives (family, team, couple, etc.), tend to
be plural when the component members are considered separately and
singular when the group is handled as a unit
So:
1) "My family is great."
or
2) "My family are great."
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.
Best Answer
I would not use the first sentence. It's fine in spoken English, but in written/formal English (especially a job application) it should be avoided.
The second sentence is fine, put I would not recommend it either. Instead, I would say:
This sounds better, matches the tenses, and get's rid of the rather vague "me." (Just saying "me" doesn't clearly state what you are asking.)