Another method that works even in ancient versions (eg 5.1) is to mimic the LATERAL join with what I call "poor man's LATERAL JOIN / OUTER APPLY".

This is the LATERAL join we want to mimic:

SELECT i.*, si.child, si.parent_id 
FROM items AS i 
    LEFT JOIN LATERAL
      ( SELECT s.*
        FROM sub_items AS s
        WHERE s.parent_id = i.id 
        ORDER BY s.id             -- assuming we don't want arbitrary results
        LIMIT 2
      ) AS si
    ON TRUE ;     -- this is redundant: ON i.id = si.parent_id ;

The poor man's version:

SELECT i.*, si.child, si.parent_id 
FROM items AS i 
    LEFT JOIN sub_items AS si
    ON  si.parent_id = i.id
    AND si.id <=                     -- 1. three places where 
      COALESCE(
      ( SELECT s.id                  -- 2. same column is used 
        FROM sub_items AS s
        WHERE s.parent_id = i.id 
        ORDER BY s.id                -- 3. as this ORDER BY 
        
        LIMIT 1 OFFSET 1             -- pick the 2nd.
                                     -- if you want LIMIT 5, 
                                     -- you need LIMIT 1 OFFSET 4 here
      ), 
      4294967295)                    -- this would depend on the type
                                     -- of the column used in ORDER BY
                                     -- and on ASC/DESC
 ;

Test at dbfiddle.uk

Disadvantages:

• complicated to write (and read/understand), especially when the LIMIT wanted is 2 or larger.

Advantages:

• works everywhere - even in ancient versions.
• adequate performance, when the number of distinct items (the parent table is small).