I have said in a previous post to Robert that it is of little use to forum readers pointing them to a website when the person posting obviously has no idea of how accurate or relevant it is.
The OP has asked for horse riding close to Fañabe and has been offered a list of 11 places - 8 of these are in the North of the island. One in the south is a golf club and not horse riding although I understand why it is included (it had stables on-site which is where Live Arico the dog rescue stayed for a while earlier this year) of the remaining two the Rancho Bonanza is approx 35km from where the OP is and the second, although I have not heard of it is within a reasonable distance although I must admit I had thought that this was the place that had moved to Alcala. I would also suggest 10 Km from resort in Aldea Blanca a typical Spanish farm where you will receive first class horse and personal service . They breed Spanish horses and have a collection of rabbits, goats, sheep, ducks, chickens and dogs. Tel: 646 807 234.