I consider that to be very much an individual decision, and I expect, as you say, that the level to which a person's bisexuality is openly demonstrated may play a factor. Regardless, I believe coming out as bisexual to your partner to be pretty important in terms or trust, respect, and intimacy.
For myself, my parents and sister have always been well aware of my bisexuality, since I never took any pains to hide it. I have noticed, as an amusing side note, that they seem to believe I've "turned" heterosexual, since I married a man and had children. So silly.
For my husband, his bisexuality is not a subject he discusses with his parents (though I suspect they know). At any rate, we knew these things about each other long before we were ever married, and it's made for some fun times over the years. We don't shout it from the rooftops, true. We keep our private lives private. But we also strive to help our children understand that it's not the gender (or sexuality) of the person you love that matters, but how their soul touches yours.