Yuri was actually Bridget's first fiancé (it was broken off by their families on account of her only having a contract with a micro-spirit). They barely knew eachother, but Yuri was present when her father burnt her hand (so trauma on his end too). Not that she remembers until she's already fallen for him (and he for her). The effect of their having known eachother is mostly a huge amount of guilt and feeling helpless on Yuri's part, which Joseph milks for all it's worth.
As for her family, she never makes up with her father, nor does she want to — the Freudian excuse is no excuse, she's well aware.
Her mother is another matter, however, and she has never forgiven herself for not stopping her husband from burning Bridget. Moreover, Bridget misunderstood what she was saying (Japanese is a
pro-drop language, so much so that her mother never actually stated what was bad or who was at fault — she was blaming herself, not Bridget). Additionally, her father never let her mother near Bridget after the incident, no matter how much her mother begged to see her, which quickly resulted in her becoming seriously depressed and developing ties with a malicious spirit (which is deadly), so that her mother could at least see Bridget in her dreams. Her father reacts to this in the worst way possible from a mental health-standpoint, which is to confine Bridget's mother to her room. On better days, she talks to the butler or her adoptive son, mostly about Bridget being such a wonderful, intelligent girl and such; on worse days, confusing dreams with reality, she refuses to let the servants light any fires, because Bridget is afraid of fire.
Bridget's adoptive brother is completely innocent and admires her (with a bit of mild sis-con in the mix; nothing creepy or romantic). The only reason he stayed away from her was to avoid making her situation worse. Additionally, while many people think that he nearly calls her "Red Fairy," he's actually just shyly trying to call her "Ane-sama" (a formal way to address one's elder sister in Japanese) — he can rarely get past the first syllable (which is the same for both, in Japanese). Later on, he helps her out quite a bit.
The web novel's resolution towards her parents was somewhat confusing — possibly giving the mother the same punishment as the father, despite the obvious differences in their actions and Bridget thinking that she wants to patch things up with her mother (not, as I said before, her father). Whether she does or not was left unresolved at the point the web novel left off (I hope the light novel and manga are better here; there are a few differences, so…). Her bother, on the other hand, she becomes close to.