It's not just about becoming bilingual, a population becoming bilingual in a "prestige language" is the first stage of language death (though of course it doesn't /always/ lead to language death).
It is true that social pressures kill off local languages, but it's usually not because parents don't want to teach their children their mother tongue, it's that people stop using the language to communicate because of the influence of the "prestige language". My parents (and all of the parents in the immigrant community I live in) went through great pains to teach their children their native language.
The burden is not in knowing another language, the burden is in making the "less useful" language your main one.
Your parents and all of the parents in the immigrant community you live in are very happy that English is now your main language which you acquired through school, friends, tv etc. Or maybe they put you in your-language-only schools, made sure you socialized with your-language-only friends and watched your-language-only media?
How would you feel if your kids only learned Chinese[1], and not a word of English?
[1] I’m assuming you’re not Chinese
Would you feel the same way if that includes all languages you know, including all versions of English?