In this paper, we introduce a novel Argument Mining task based on the existing task of Argument Structure Parsing (ASP). Our new task, which we call ASG Parsing, is the task of generating Argument Summary Graphs (ASGs) from dialogical argumentative text. We release a dataset containing ASGs, a type of graphical summary for argumentative dialogues, in which the nodes are summaries of statements and the edges are the argumentative relations between them (support or attack). We approach the problem with two different LLM-based solutions: (a) a pipeline system involving two models separately fine-tuned for summarisation and stance detection; and (b) an end-to-end system based on the TANL (Translation between Augmented Natural Languages) framework [1]. We show that the TANL approach outperforms the pipeline approach across the board. We also show that, for all systems, performance degrades as the depth of the graphs increases.