Be fair to her (and the potential dog). If you have reservations, something is missing - then it's missing.
I watched "He's just not into you" last night.

My opinion is that you let her go.
And it isn't fair to have a dog, especially a puppy, if no one is home for most of the day. They really are like perpetual 2 year-olds - they suffer when left alone. And your parents are not enthused...

Is it fair to start off relationships where one of the parties are not loved fully?

Of course, I could be talking out of my.....
(((hugs)))