At present, it is possible to open an Issue and see "Child issues" in a List View. It is possible to perform in-line editing of field data for some fields, but not for fields such as "Start date". 

This Suggestion requests the ability to edit "Start date" field data in-line.

Workaround

In the meantime, there is a workaround you can use to inline-edit both Start date and Due date when working with multiple subtasks:

1. Switch to the List view, and expand the subtasks under the parent, or

1. Use an issue search (All work / Issue navigator) with a JQL filter like: project = TES AND parent = "parentworkitem" ORDER BY created DESC 

From there, you should be able to inline edit both the Start date and Due date fields for those issues.