One of the imagined advantages of being an architect rather than, say, an accountant is that one can arrange one's life free of bourgeois conventions about appropriate dress, premises, table manners etc. In practice, this just means that architects wear velcro-fastened shoes, work in offices which do not have air-conditioning and have to put up with the boss' cretinous spaniel slobbering helplessly at their feet.
As office pets go, however, dogs are the appropriate choice. A glossy labrador curled up on a Barbour jacket in the back of your Volvo is all the reassurance your client needs that yes, you actually do know about lead detailing and thatch; it is easier than learning what words such as 'wainscot' mean. In town, a polite fox terrier wearing a Burberry dogcoat might just convince a sceptical client of your status as an arbiter of taste.
Architects have no instinctive fondness for cats.
Cats show little sign of needing people, and architects need to feel needed.
Also, think of the design of catflaps and litter trays; there aren't brushed stainless steel versions of either on the market, so architects can't be interested.
Parrots are rare, possibly because of their potential for inappropriate comment.
Tortoises are even thinner on the ground. Being slow and unfashionable, tortoises would have been a more fitting companion in the 1970s.
Goldfish had their moment in the limelight, the combination of bright colours, excessive use of glass, and not having a point of view about anything much appealed to architects' sensibilities. The vogue faded, however, with the realisation that offices with fish tanks attract dopey staff and are susceptible to structural failure and condensation.