I think, you could try first; if it is only the rate factor that is affecting the decision of parents. You've also mentioned you do not have a big back yard; but there are beautiful parks to compensate that. As far as you take the children out everyday. And people sees also that, if you do outings such as visiting a library or playgroup etc. When parents see you out is what sells most. I do not advertise I meet mostly new parents in my outings. I go out everyday, during the mornings (weather conditions permiting).
I see some of my friends in the same situation empty spots and especially when September comes our older children start school and we need to fill those spots.
Lately I see, more and more caregivers looking to fill their spots, luckily we have not gonne lowering prices yet.
Tell your prospective clients that if they're interested you might negotiate your rate. And see if they take the spot or get back to you.
Also my other suggestion is to take pictures of your group on your outings so, parents can see you do things disregards a back yard or not.
I hope it helps.
If lowering your price is a determinant factor in the desicion making process then it might benefit you, if not then it is the combination of non back yard and higher cost and the opening of more spots in the licensed sector.