An advantage of parliamentary models is that you don't have to know the utility functions of the individual agents, but can just use them as black boxes that output decisions. This is useful for handling moral uncertainty when you don't know how to encode all the ethical theories you're uncertain about as utility functions over the same ontology.

Do the choices of a parliament reduce to maximizing a weighted sum of utilities?

Let's say the parliament makes a Pareto optimal choice, in which case that choice is also made by maximizing some weighted sum of utilities (putting aside the coin flip issue). But the parliament doesn't reduce to maximizing that weighted sum of utilities, because the computation being done is likely very different. Saying that every method of making Pareto optimal choices reduces to maximizing a weighted sum of utilities would be like saying that every computation that outputs an integer greater than 1 reduces to multiplying a set of prime numbers.