Lazarus is laid at the rich man's gate, covered with sores and longing for food from the rich man's table. The gate is more than scenery. It marks the boundary between abundance inside and misery immediately outside.
The rich man does not need to search the city to discover someone in need; Lazarus is at his own entrance.
Large private houses and estates could have substantial gates controlling access and displaying the status of the household. Beggars and dependent people naturally sought places where wealthy residents and guests might pass.
Lazarus' condition suggests extreme dependence: he is described as having been laid there, rather than simply choosing a convenient begging location. His desire for what falls from the table intensifies the contrast with daily luxury inside.
The architecture turns social distance into physical distance measured in only a few steps. Wealth and suffering coexist on opposite sides of a gate.
That makes the later reversal in the parable especially pointed. The rich man who failed to bridge the small earthly distance to Lazarus encounters a great fixed gulf that cannot be crossed after death.