Byways are open to horses and yes, they are marked with red arrows the same as bridleways are marked with blue.

Can horses ride on byways?

A footpath is defined as a highway 'over which the public has a right of way on foot only', so horse riders are restricted to bridleways and byways. ... People keeping horses must ensure that their animals don't cause a danger to other people.

What vehicles can use a byway?

. A restricted byway allows a right of way on foot, on horseback, or leading a horse, cycling and for any vehicles other than mechanically propelled vehicles. In some cases there may also be a right to drive animals.