What were a parking rules for horsemen in big cities in 19th century?

It is a good question. The Carter seemed to have unloaded his wares in front of the shop and the mover in front of the house, according to photos  (though they may have been posed there to show off the name on the cart and the photos used on advertising cards) Where did the coachman go after he dropped off the master and his party in front of the Metropolitan Opera house or Covent Garden Theatre. Or the White House or Buckingham Palace?

When the Mackay Bennett brought the bodies of Titanic sinking victims to Halifax, there was a photo of a line of horse drawn hearses or flatbed  wagons at the pier -- one beside the next, the horses facing the pier; one long low black coffin on the back of each wagon, one or both men at the side of each horse's bridle. IMO it was  one of the most somber pictures related to a disaster I ever saw.

That would depend on where the big city was and for how long you were leaving your horse.

If you just tied it to a convenient railing in the biggest city in the world you would come back to find an empty space.

You didn't really 'park' them and leave them as you would a car. You kept them in stables which were located in the streets behind the big houses (in London you still find them - that's what mews houses were built as and then later converted).  Our coal man had one and the horse wandered along happily with a nosebag on. Nobody worried too much about what came out the other end, it made marvellous compost for the roses in town gardens. Horses for Hansom cans could wait on taxi stands in the day time and at night were turned out into the fields that were still close to city boundaries. The rules were more about controlling the horses to avoid them running amok - as my uncle's did the night his city was hit by an air raid, but that's another story.

This is a very good question. One important rule might have been, "Curb your horse." Just imagine the size of plastic bag you'd need to clean up after a horse.