Well, they likely died when he was a baby, that picture is obviously very old and Obito only looks a few months old, he may have been born either at the end of the 2nd War or during the very beginning of the 3rd.
Also, it wouldn't have been Obito's fault, if anything he was shown to be an incredibly kind child, not unlike Naruto, but, since he wasn't "cool" or "skilled" he was invisible, and, take this from a person that's been invisible, it sucks, knowing no one cares.
We don't know that Minato and Kushina always cared, we know they saw him as a son once Minato became his teacher, but before that I'd imagine Rin was the only one. If you meant me saying Minato didn't care, he did, Obito likely thought he did, as his berating of him seems to imply that he wanted Minato to stop him.
And, you still don't have evidence of this, Naruto was filled with rage and being charged when he learned of Kakashi's fate, and he didn't even see him die. Obito, on the other hand, Naruto saw his entire life, this man could've been akin to an older brother had he not turned (given his dreams and his relationship with Minato and Kushina). I think Naruto was partly sad and proud, proud to know that Obito died as a shinobi he'd aspire to, that after all his pain, he moved past it, and found peace, even in death. I honestly think Naruto may have been more moved than sad, like, I've cried over things that moved me, what could be more moving than watching your once bitter enemy, truly a sad and lonely man, not only redeem himself, but to give his very life to protect you? Pretty damn moving, I'd say.
