With regards to English Language and Linguistics as a subject...
Personally, I qualified to take H2 ELL at numerous, or practically all the Junior Colleges offering the subject (double distinctions for English and Pure English Literature for my O' Levels) - but frankly, I just didn't have the passion or patience for what I found to be a rather "dry" subject.
As for your choices at this current juncture of your life...
I can't give you any "professional advice", but just my honest opinion.
I'd say that your command of the English language is clearly quite superb- and if you were an A' Level student (private or otherwise), you'd have a clear advantage over others in Arts subjects such as History, General Paper and Literature.
However, as to whether you ought to take the A' Levels as a private candidate or not, that is something you should really think hard about. Your parents are right- what matters is that your well-being and future livelihood is taken care of. Some may say that since you're already on the path to getting an engineering diploma/degree, it'd be silly to alter course now, while others may say otherwise.
You *could* go by the
A' Levels --> NUS Political Science (something that may interest you?)
or
A' Levels --> NUS Faculty of Arts & Social Sciences
route if you choose to- but know that it is ridiculously competitive to get into the university via any medium, be it a diploma or A' level certificate. And especially for private candidates- without the benefit of peer tutoring, teachers' aid and school resources, it is even more difficult for you.
Passion alone doesn't get you too far, and neither does innate talent. You need a combination of hardwork, aptitude and attitude to achieve stellar results academically.
But also- you're my age right now. We're young, and since you have supportive parents- talk it out with them. I'd personally be inclined to follow my dreams, if i were you. But discussing with people more mature than ourselves is usually a good idea to gain more perspective before making a decision.
All the best!