Hmm. I wonder whether they doubted that the low fence was at all theirs? In a way, their choice to build a new fence rather than adapt the one that was there already might suggest that they feel the legal boundary is between the two fences.
In practice, it is quite unusual for a fence to be of shared ownership. Usually it forms a physical boundary, but the legal boundary is on one side of the fence. This means the fence will be owned entirely by whoever's land it is on, and the owner can maintain, neglect, or even remove it as s/he feels fit.
With normal people, you could just ask them to write a letter saying they do not own the low fence, as you want to replace it and do not want a dispute should one of you later sell up.
If you think they are going to be difficult, I'd be inclined to write to them asking for money towards painting the existing low fence, unless, of course they feel it is yours and nothing to do with them, and in which case you will do it at your own expense.
Hopefully they will fob you off with a reply that they own the high fence and the low one is entirely yours and not their responsibility so they won't pay a penny. This is, of course, the outcome you were hoping for. This would make it very hard for them to later claim they owned the fence you wish to remove and should prevent them from being able to take any form of legal action - after all, you will have removed a fence they quite clearly told you they do not own!