When a parent dies with no living spouse, there are two children, who has the legal authority to finalize funeral arrangements?

If you were to open an estate, the children would have the say and would either agree on which one is the Executor, agree that both are Co-Executors or fight it out in court. I think the funeral home wants the children to agree in such a way. If not, especially if there is a feud among children, the funeral home will not want to deal with you. For example if one child is demanding cremation for mom and one a traditional Christian burial with 3 day viewings, the funeral home may tell you to go to the next funeral home down the street. The business of dead people is a steady flow, and they want to avoid problems.