A: Just concatenate the table twice. Once that's done, you'll have to use the table alias:
SELECT a.Parent_ID,bp.Name AS Parent_Name,a.Child_ID,bc.Name AS Child_Name FROM TableA AS a INNER JOIN TableB AS bp ON bp.ID = a.Parent_ID INNER JOIN TableB AS bc ON bc.ID = a.Child_ID
Note that TableB appears twice in this query. This means that we must use the alias every time it occurs, and qualify the columns with the appropriate alias in the SELECT statement. Since the columns in the two instances of TableB have the same name, the name is common but not required - so we use the SELECT statement to duplicate the table alias.