"My parents' yard has a lot of trees."
Translation:El jardín de mis padres tiene muchos árboles.
Your parents own this garden. "A mis padres" is not possessive; "de mis padres" is.
Is there a better way of writing "My parent's yard" than "El jardin de mis padres", perhaps using "su"? Like "Su jardin".
Probably not, since it's a different sentence.
"There are a lot of tree in my parent's yard."
"My parent's yard has a lot of trees."