When branching off of the basic process, task items display on the sprint board. However bug items (and their child tasks) do not show their parent bug. See Below:
Tasks should show their parent. In fact, if I edit a bug from the board, I can see it on the board. See below:
Unfortunately, after refreshing hte page, the item returns to the unparented category.
TL;DR - How can I configure my sprint board to display task parent items rather than only features?
edit: Gear image 
On
View Bugs on Sprint Tracking Board
The reason for this issue is that your basic process limits the bug workitem type.
When you check the Backlog levels of the basic process：
Obviously, the custom Bug work items appear on the requirement backlog or the iteration backlog based on the team's setting. However, there is no such team's setting about it for the basic process.
To resolve this issue, we need to change the basic process to other process which track bugs on the Backlog, like: Agile, Scrum.
You could check all the process on this document:
After change the basic process to other process, we just need select the Bugs are managed with tasks on the settings:
Since the Basic Process does not have a Bug type out of the box, the Board/Sprint views do not support customizing those types at all. Since you make a fork of a basic process, you're missing that configuration option as well.
To manage bugs as you are describing you would need to switch the project to one of the out of the box Processes that support Bugs as a type or create a fork of one of those processes.