[edwinghansen]

Hi,
I am not sure I understand what is being asked. But you have nominated " The old walk home", as a sentence, therefore it has a subject and a predicate, which of course must include a verb. So initially the sentence needs analysing. I see the subject as 'The old' the verb as 'walk' and 'home' as the adverbial clause of place. The next step is then to dissect the analysied sentence into parts of speech. Determiner, noun, verb, (to) noun.
If "The old walk home" is not a sentence then it is a collection of words and every part of speech that can be attributed to each single word is the correct answer. ie if "old' is an adjective to the noun "walk" then we do not have a sentence, as you nominated in your introduction.
Very interesting
edwin.