I asked my landlord if I could keep a small dog in the apartment for therapeutic reasons. I struggle with depression, anxiety and ptsd and felt a dog could provide emotional support.
I assume "property management" works on behalf of the freeholder or a group of leaseholders and the landlord is a leaseholder - otherwise it would ultimately be the landlord's decision alone.
It's common for leasehold agreements to prohibit pets unless there is consent.
The terms of the lease are important.
If there is nothing about pets, then they can't prohibit pets on the basis of the lease.
If it simply says something to the effect of "no pets allowed, end of discussion" then that's automatically unfair therefore void (whether that means keeping a pet will be stress free is another story).
If it says something to the effect of "no pets allowed that cause a nuisance" then a pet is allowed unless / until it causes a nuisance.
Perhaps it says "No bird, animal or reptile shall be kept", in which case you could keep some fish.
Suppose it says "no pets allowed without consent" - consent must not be unreasonably refused. What reasonable means depends on the circumstances but the majority of other leaseholders refusing consent is likely to be reasonable refusal. So what did you tell them, what did they say is the reason for refusal and are they open to negotiation? You know, did you say "I'd like to keep a dog" or "my doctor says that keeping a dog will help me with my depression, anxiety and ptsd"? (I empathise that you may not want to reveal such things but it might be helpful to you if you haven't already.)
Also this specific clause in the lease could be changed - although that would likely cost say £150-300 for a 'deed of variation' (assuming all parties agreed).
Unlike in some other jurisdictions, in the UK there is no legal recognition of emotional support animals, so you can't point to a law that says you have a right to keep such a pet. However, leaseholders, freeholders or property managers might be persuaded to take the stated need for such an animal into account when they consider consent or changing the lease.
Someone might say "just move" but this is a common prohibition so moving won't help unless you know in advance the new landlord / potential landlord consented to keeping a pet and was free to do so (it is unwise to rely on the letting agent or estate agent to tell you whether pets are allowed).
Victory Place Management Ltd vs Florian Kuehn and Marielle Kuehn is a relatively recent judgment that discusses some of the above issues (summary).
Summary: if it's a leasehold property the lease likely prohibits pets, you might be able to negotiate obtaining consent to keep an emotional support pet but you have no legal entitlement to keep such a pet in that property.