That's an easy answer -if they were taught English by American missionaries, they also picked up their accent. I love Dr. Rey, can't help it!!!
OK, removing myself completely from my dislike of Roberto, and even setting aside that as he tells the story, the missionaries rescued him from a neglectful, seldom-present father, thus making it unlikely that Rey Sr was faithfully attending English classes, and with the qualifier that just about anything is possible, and even the most likely things have happened somewhere, sometime, to somebody, it is about as close to impossible as we can get that anyone, in any class taught by anyone else, is going to actually become
fluent in another language without day-to-day regluar life exposure that would be extremely difficult and unlikely to be present in an environment where a language other than the target is predominant.
Even someone with an extremely high natural flair for language, and an exceptional talent for mimickry - and mimickry is what we do when we practice pronunciation in a language we are learning - is not going to achieve the same result as someone with even a lower natural ability for language learning who lives for a time in an environment where the target language is heard almost exclusively. And the "time" for most people would be in terms of years.
Even then, it is not unusual for someone to live for decades in, for example, the United States, study, work, raise a family, and still retain to some degree, an "accent."
The English proficiency of Rey Senior is that of a native speaker, and the accent he has is an American regional accent, some elements of which might be picked up in a class, but whatever the true story of the Rey family, missionaries, or anything else, including Rey Senior's language-learning abilities, or the teaching methods of the missionaries, this is a man who at the very least, spent a considerable number of years, most probably starting when he was very young, living in the United States, and hearing and speaking all English all the time.
It would be interesting to hear with just what accent he speaks Portuguese. Unfortunately, the show did not give me a chance to hear that, but Rey senior is either a native speaker of English, or so very nearly so that the difference in fluency is imperceptible, and that is not something that one comes by even going to very good classes for a long time and studying and practicing, even watching TV and listening to music in the target language exclusively - the best language-learning method - will not produce that result.
And I will re-emphasize that my opinion on Rey Sr's English would be exactly the same even if I really liked Rey Jr, or even if I had never seen the show, and saw only that part of that episode.