A blanket "no pets" clause is unenforceable, and would remove even the requirement for you to ask before keeping pets (ref:landlordlawblog). This is why the clause contract has the phrase "the Landlord’s consent [...] will not be unreasonably withheld". You shouldn't take this phrasing to mean that the landlord is happy for pets given certain conditions; rather the landlord doesn't want pets, but has used the letting agent's standard contract which has been written with advice from lawyers.
Withholding consent with no explanation is unreasonable (see landlordlawblog ref above). The landlord simply wanting no pets, in and of itself, is almost certainly unreasonable. Given that you proposed damage repairs and cleaning, I am personally convinced that the rejection is unreasonable.
That said, the landlord might be able to reasonably withhold consent on noise grounds (parrots can annoy neighbours!), but can only do so by stating that to you (which he hasn't done).
(Im)practical answer 1
You can try fighting it out in the courts if you like. You could well win, but this will cost you money (and probably a lot of stress) and antagonise your landlord.
The landlord will then want to get rid of you (both because you're doing something he doesn't want to the property, and because you took him to court). He'll want to get rid of you with the minimum notice possible; if you're in a fixed term, that's the end of the fixed term; if you're in a periodic tenancy it's probably 2 months.
In theory landlords can't carry out retaliatory evictions. But your only chance of fighting a retaliatory eviction is... the courts! Cue more cost and stress. And it's very possible that the landlord might successfully come up with another reason to justify the eviction, e.g. "I want to sell the property", "I want to provide my nephew with a home", or if he tries putting the rent up and you refuse "I can get more rental income by re-letting the property".
(Im)practical answer 2
You could just get the parrot anyway. The landlord won't find out until his next inspection, and even then will find it difficult to evict you in the middle of a fixed term for such a minor breach of contract... but that doesn't stop him evicting you at the earliest possible "no reason" eviction date. This would again be either the end of the fixed term or 2 months if you're on a periodic tenancy. And you wouldn't be able to fight this as a "retaliatory eviction", as there will be no registered disagreement between you.
Your legal rights and your practical options aren't the same thing, unfortunately. Your best option is probably to give up on the idea of getting a parrot.