I'm not the best support player, but maybe I can give you another point of view.
Zyra doesn't have any ability to heal her lane partner, nor does she have an escape mechanism. Additionally, her harass is based on her skills.
To counter her, I would recommend Soraka, for 2 reasons:
Since you have the aability to heal your carry and Zyra doesn't, you can stay longer in the lane and harass the opponent. Ask your carry to harass as much as possible, and you will force them to go back to base, giving your partner a gold advantage.
Zyra is based on her skills; as Soraka, you have the ability to silence her every time she gets close to you or your partner.
If Zyra is still bothering you, simply put a ward in her brush. This allows you to predict her moves and gives vision of her to your carry, so he can harass her.
If I could choose a lane partner, I would pick Caitlyn, as her long range makes her pretty good at harassing the opponents.
As a support, your role is to keep your carry safe. If Zyra tries to harass you, silence her and attack back, healing right after. Soon, they will have low hp and your hp will stay full.