How can I update the "status" field in the parent table when there are multiple records with different statuses in the child table, such as "Started," "Inprogress," and "Hold"?

You can make a summary field in the relationship. Click the box that says 'Whether any records exist' for your summary field, and add the filter 'Where status = On Hold'. So now you have a checkbox on the parent that tells you whether any children exist that are on hold.

If the Status on your parent is a formula field, then you simply just say If the checkbox = true, the status is 'On Hold', else the status is 'In Progress'.

If the Status on the parent is a manual entry field, you will need to use a pipeline. Trigger it off the child table and say, when the status of a child is changed, search for the parent and if the checkbox is true, set the parent status to on hold. If not, set it to in progress.