The issue isn't whether teen you can look after four birds. Trust me, this is probably the period of your life when you have most free time and fewest responsibilities.
The issue is what happens after you leave school. If you're planning to go to university, it will be difficult to find accommodation that will let you keep your birds. University schedules are demanding: you will have lectures, seminars, study groups and, depending on subject, lab hours. You may be offered opportunities to go on study trips, or even to study abroad for a semester or academic year. There are also lots of social activities and student groups that you might want to be involved in.
I have no idea what kind of career you want to have, but many careers involve toughing out a few years of instability before you get more established - perhaps long hours, a commute or changing shifts, including weekends, or a combination of these. Again, you may be offered opportunities like moving to a different location to get a promotion or further training. You may want to try a couple of different fields to see where your talents lie. All this disruption is something that a human can take in their stride, but will make a bird anxious at best and mean that you have to rehome it at worst.
You may want to move around between regions and even countries. Finding accommodation that is pet-friendly can be tough; many leases have a "no pet" clause and finding accommodation with multiple pets is even harder. There are lots of restrictions on travelling between country with a pet, so if you want to spend time in another country then you need to work out who will care for your bird(s).
You may want a relationship. This will mean more time away from your birds so you can spend time with a partner. Your partner may be otherwise perfect but hate birds.
Birds aren't like other pets. They thrive on routines and attention. They are not happy with an hour here and an hour there, at different times of the day. As
@Sodapop&Co. says, sometimes they will have relatively low maintenance days and sometimes they will be causing havoc every time your back is turned. They need at least 3-4 hours a day out of their cage. Unless you have an actual aviary, no cage is big enough to allow them to fly and exercise.
When they get ill, it can take hundreds of dollars to even identify what is wrong - let alone treat it. Your parents may balk at $400 for just tests.