If you are going to translate household chores and babysitting into money for riding lessons, then you have to set a value for each chore. I.e., cleaning the bathroom equals X dollars.
You also have to set up expectations of what is clean - a 13 year old's version of clean may not be your version of clean - in short, you will have to teach her to complete the task first to avoid any miscommunication on what the "cleanliness" expectations are. Same with babysitting - what are the expectations. When watching her sister swim does that mean she gets to play an electronic game or does that mean she has to be in the pool actively engaged with her sister.
Set a list of chores that she can reasonably handle, along with a time frame for completion. Again, i.e., bathroom to be cleaned every Wednesday. Go over the list with her, give her time to grouse and ask questions - in short, you are making a contract with her completing tasks and you "paying" her for those tasks.
If she comes up with additional ways to earn money, by all means let her approach you with the ideas and give them serious consideration. As that would be, to me, an indication of her willingness to "earn" her riding lessons.
For me, with my soon to be 17 year old son, he began assisting with household chores, in seriousness, when he was 12. We made a list of chores that included who did what and when the chores should be done (daily, weekly) and posted it on the refrigerator. As he has gotten older he has taken on more responsibilities around the house, especially now that I work 1.5 jobs. I do not pay him for chores - other than lawn mowing, and even that is nominal. Assisting with household chores is just part of being a family member - plus teaching him life skills.
I get that she seems to take what you give her for granted. My son did also. Until my finances drastically plunged 3 years ago. While he always had a vague idea of what I earned and what our expenses were, he paid no heed to it, because we always had money. When that changed, it was time for me to explain our finances in more detail to him. It was time for him to understand that the extras we had were going to be curtailed because the mortgage and electricity are more important. It was a painful lesson (for us both) but in the end I have a teen who thinks before he asks for something and is willing to compromise in order to stay within our budget. You may try the fiscal responsibility talk with your daughter also, as a way to explain why you feel she now has to "earn" her lessons.
Also, teens are pretty self centered - hence the grouchiness and lack of thanks. Sigh - it passes.