I'm a bit confused. 'paard' is a 'het' but the adjective has a final 'e' this definite article het (Het + adjective with final e + noun)
with indefinite 'een' the adjective has no final 'e' (Een + adjective without final e + noun)
In the context of this sentence 'Het oude.....' means 'The old....'. On its own, 'Het oude' means 'The old one'. In any language, context can cause that identical phrases can indeed mean different things.