If they are parrying you on wake up, stop trying to attack them on wake-up. Best course of action is like ToyRobotTerror said. Get next to them on wake-up and just back out of throw range and block until they are fully up.
As Chun Li you can poke with c.MK against because of it’s speed and range, it’s probably the safest poke you have, though your opponent could probably SA3 after parrying you if they are competent enough. s.RH pretty fast and high priority as well, but see what works best for you. Learn to Super-Jump-Cancel any moves that allow it as having mobility to stay out of Ken’s DP and Throw range will help a little. Keeping away from corners shouldn’t be much of a problem if you ever back yourself into one, wall jump etc. Don’t forget to utilize your Air Throw against jump happy players.
Most of it is really up to your ability to cope with the opponent and how well you are at managing your distance against the opponent. Learning to Kara Throw could be a great asset for you if you don’t already know how to, it keeps pressure on them and pressure can crack certain people making them do stupid things sometimes.
Against Ryu and Akuma there are different worries (in general) so you’d have to change certain aspects of your game for them, but generally, you want to mix it up with different normals and throws. Try not too readable to the enemy, even though that’s easier said than done. Just look at how you play and if you’re doing the same thing over and over in certain situations, it’s time to learn something new. For example, b.HP on wake-up everytime will get you caught so change it up. Back out of throw range and UOH / c.MK / s.RH / close s.RH SJC jump away. I’m not much of a Chun player so I can’t give much advice, but take what you can from my post and see what works for you.