I think its safe to say that the Old World birds are extremely different from the New World birds.
Thus, I would not (no offense to anyone) take the advice of someone owning a Macaw, Conure or Amazon etc (New World) with regards to the ability of a Senegal, African Grey or etc (Old World) to share their humans.
Old World birds flock in single-species and thus seem to have a very well defined sense of 'us' and 'them'. New World birds are often seen in large mixed-species groups and thus seem more accepting of 'outsiders'.
My Timneh Grey refuses to live with another bird and will attack/provoke/have a jealous plucking fit on sight of another parrot if its on familiar turf. She bit and caused nerve damage to both my ex-tiel's feet, tore out his toenail through the cage bars and ripped out tailfeathers. Whenever he was out, she would go immediately to her cage lock and try to chew it off. The cockatiel long-term was always sickly (partially due to a long battle with giardia, but also due to stress and anxiety) and it was rare for me not to have him on anti-biotics or have to watch his droppings which were never normal. He had extremely loud stress and fear based screaming. Ultimately he was placed in a rescue and is now much happier
This seems to extend to non-avian pets as well. My TAG will also shriek at (its her intruder/defense call) and dive bomb or chase cats, dogs and other small creatures even if they're not bothering her or her cage. Thus, I would not, ever, ever add a predator species to the home as she would go after them and lose ultimately when the cat/dog/snake etc defended itself.
I also agree that when you have two highly social but also highly incompatible companion animals in the home that one of them loses. One becomes the baby and gets all the attention the other becomes the nuisance and the one who gets the short end of the stick. The one who 'if only we didn't have it then the bird/cat/dog would be happier and get more attention, not be locked away and we would be less stressed juggling schedules and be less guilty'. I experienced this with the cockatiel, I always felt bad because he always came second and I had to watch them 24/7.
I would question your motive (again, as someone who as been there) as to why you need another parrot. Often it is because the first bird/pet is not living up to your expectations, so you get another hoping that one will be 'cuddlier, smarter etc.'. What you end up with is a house full of minimally-cared for, anxious, stressed and borderline neglected animals. Or in my case you end up with three stressed out birds who don't get all they want, become more like liabilities who need to be carefully rotated out for 'playtime' instead of a cherished family member. The larger the animal collection so to speak gets the more they become display animals---hence the word collection.
I would not get a snake, because you are risking either stress or aggression on the part of the snake from the scent of the parrot. A parrot will also be very stressed by the sight of a snake, even if secured in a cage. Of course...if the snake ever gets loose...
Everyone else has told you of the cat saliva etc bacteria risk, so I won't repeat. A dog is an even bigger threat due to size and again breed differences have been mentioned already.
All this to say I would focus on putting your energy and love into your senegal. IF you must absolutely have another pet may I suggest a small lizard like a gecko? They don't need much attention and live safely in glass terrariums away from jealous beaks. Due to salmonella etc risks of course careful handwashing is a must though.
PS: Not to start a fight on an off-topic subject but Shelby I highly disagree that the Dog Whisperer's methods are cruel.