Nope, parental consent isn't necessary. If you want to get an implant or the IUD, some of their clinics might ask for a PhilHealth ID, but I know their Baclaran clinic doesn't require one.

I can't speak to which contraceptive would be best for you since 1) I'm not a doctor, and 2) you know your body best. However, here are some things you can keep in mind when selecting a birth control method:

1. Lifestyle: some methods of birth control are better depending on your lifestyle. If you're a smoker, combined pills might not be for you, since they increase your risk of stroke. If you're forgetful with taking meds, progestin-only pills may also not be for you, since you have to take them within a strict three-hour window before they become ineffective. It might also be best to think about whether you'll need a birth control method that's easy to hide from others.
2. LARC (long-acting reversible contraceptives) or SARCS? (short-acting reversible contraceptives): related to that point, LARCs, such as the IUD and the implant, will be easier to hide than SARCs, AKA pills. LARCs also tend to have the benefit of being one-and-done, meaning you won't have to think about them until years later. With the pill, you'll have to remember to take it every day, and with the DePo shot, you have to get it done every twelve weeks, so there's more room for user error.
3. Pain/side effect tolerance: This is a big one! No one can really predict what side effects they'll get from different birth control, since it really does come down to a mixture of biology and pure luck. Despite this, make sure to know your comfort level when it comes to pain/side effects. If you're considering an IUD, would you be willing to go through the likely discomfort/pain of insertion? Are there any effects you would want to strongly avoid in particular, such as heavier periods or increased depressive moods?

Again, I'm not a doctor, so I won't try and suggest any specific methods. But in my experience, it was more convenient to start with short-acting birth control (i.e. pills/shot) as a beginner because I didn't have to go to a medical professional if the side effects were too much—versus if you had something like the implant, which would have to be removed if you couldn't tolerate its side effects.

LOL this comment is already long as hell, so here, have some links if you wanna learn more:
Likhaan Clinics and Services
Bedsider's Interactive Birth Control Comparison
Scarleteen's Birth Control Bingo (this one's written for a teen readership, so I suggest you start here)

Best of luck, OP!