The duty
A tenant can request to keep a pet and you have to respond within 28 days. You can refuse, but the refusal has to be reasonable and it has to be given.
What "reasonable" is not
A blanket no. A policy of refusing every request regardless of the pet, the property or the tenant is the thing the rule exists to stop.
The clock
Log the request when it arrives and the 28 days runs from there, visibly. The most expensive outcome here is not a refusal that gets challenged, it is a request nobody answered.
Records
The request, your decision and the date it was sent are all recorded, so the position months later is a matter of record rather than recollection.