Actually, now that Mouthwash mentioned parents, I guess I could mention mine as well. My grandparents also learned to speak fluent English too, though I am not entirely aware of how that happened - I do know that they were typical of the upper-class, Western-educated intellectuals so it was probably a combination of schooling (particularly abroad or at international schools) and interacting with English-speaking folks often.
Anyways, for my mother, I'm less certain about. I do know when she came to America at ~13 years of age her English was heavily accented; however it does appear she knew a competent amount of English as it does not appear she had much difficulty adjusting to school (relatively speaking). It probably also helped that my grandma was formerly a principal of an English-speaking international school in Vietnam (and that my grandma also earned a college degree in America decades before). It appears that anyhow her English became less and less accented over the years simply due to exposure; the fact that my mom's side lived on the east coast of the US, where there were fewer Vietnamese, may have helped.
Anyhow, as to my father, he was first introduced to English by my grandfather (who, as hinted above, probably learned it as part of his schooling). My dad told me - quite fondly too - how one of the few times my grandfather, who was otherwise your typical strict and distant Confucian parent, would do anything with him father-son was when he taught him English. Most of this English-teaching (and French-teaching, apparently) happened while my grandfather was working in Japan as the spokesperson (or aide/assistant, I forgot) for the S. Vietnamese ambassador to Japan. I believe these lessons in English were reinforced due to the fact that my dad went to an English-speaking international school while he was in Japan, so he would have to use English more frequently than just during his lessons with my grandfather.
Thus when my dad went to America, at ~14 years of age he already had a decent command of the language, I'm assuming. He came a few years before the Fall of Saigon, when there were few Vietnamese in America, so this probably also helped when it came to learning English as he was forced to deal and speak with English-speaking Americans - from a wide variety of backgrounds, not to mention, including whites, blacks, hispanics, and non-Vietnamese Asians - on a frequent basis at work and school and other places in public. My dad also appeared to enjoy watching movies and tv and music and other pop culture like other kids of the day (he really liked Scooby Doo, and the Beatles, although I guess the latter were sort of getting out of fashion by then). He also read voraciously, particularly about religion and history, which also helped him understand English in contexts other than simple greeting and light conversation. Depiste all this he apparently still spoke with a heavy accent for a number of years (possibly until right before I was born), but managed to teach himself how to speak "properly". Lastly, while my dad lived on the West Coast, he tended to live in regions in less Vietnamese, not to mention during the crucial years - i.e. when he was young - there weren't that many Vietnamese in America as the Vietnam War hadn't ended yet.
So I suppose for my parents, particularly my father, it was a combination of coming from a background where they already had some theoretical familiarity with English, and then being able to apply it in practice when they were suddenly thrust into an English-speaking world.