ROME—Former European Central Bank chief Mario Draghi was due to start talks with Italy’s divided political parties on Thursday to seek backing for a new government to deal with the entwined health and economic crises. Draghi, one of the most respected figures in Italy who has received a mandate from the head of state to put together an administration, entered parliament in mid-morning and was due to see leaders of the smaller groups later in the day. He was expected to meet the main parties on Friday, with many political leaders undecided over whether to endorse his appointment or even look to join his new cabinet. Financial markets have cheered the sudden arrival of Draghi, hopeful he can spearhead long-awaited reforms to boost growth in the euro zone’s third-largest but most sluggish economy, thereby bolstering the currency bloc. However, he needs the backing of parties from across the political spectrum …