How to find other parent table entries that have exact same values and number of child table entries?

SQL: Returning an id from a table based off the values in the child table.(SQL)(MySQL)

I have 2 tables, Parent (ID), and Child (Value_A, Value_B, ParentID), with a 1->many relationship (Parent has multiple children).

Given an Id for the Parent table, how would I find other entries in the parent table that have the same values and number of Child table entries?

Parent

Child:

Given Parent ID 1 the expected result is 5, 6.

I've had queries that return Parents with the same Value_A but not Value_B and vice versa.

This is a classic Relational Division Without Remainder question.

There are a number of solutions, but the classic one is to first get the total number rows in the original set. Then left-join that set to each potential set, and ensure that all rows match as well as there being the same number of matches as there are in the original set.

DECLARE @yourId int = 1;
DECLARE @count bigint = (SELECT COUNT(*) FROM Child WHERE ParentId = @yourId);

SELECT *
FROM Parent p
WHERE p.Id <> @yourId
  AND EXISTS (SELECT 1
    FROM Child other
    LEFT JOIN Child cThis
       ON cThis.Value_a = other.Value_a
      AND cThis.Value_b = other.Value_b
      AND cThis.ParentId = @yourId
    WHERE other.ParentId = p.Id
    HAVING COUNT_BIG(*) = @count                     -- same number of matches
       AND COUNT_BIG(*) = COUNT_BIG(cThis.ParentId)  -- all rows match
);

db<>fiddle

For a With Remainder version, just change the LEFT JOIN to INNER JOIN, and change the HAVING to

    HAVING COUNT_BIG(*) >= @count   -- at least that number of matches

To compare sets of values, can use aggregated strings and compare them.

Try this

with ChildValues as (
select  Parentid
  ,string_agg(value_a,',') within group (order by value_a) sA
  ,string_agg(value_b,',') within group (order by value_a) sB
from Child
group by ParentId
  ) 
select * 
from ChildValues t1
left join ChildValues t2 
    on t1.sA=t2.sA and t1.sB=t2.sB
    and t1.ParentId<>t2.ParentId
where t1.Parentid=1

Result

Or full query

with ChildValues as (
select  Parentid
  ,string_agg(value_a,',') within group (order by value_a) sA
  ,string_agg(value_b,',') within group (order by value_a) sB
from Child
group by ParentId
  ) 
select t1.ParentId, string_agg(t2.parentId,',') within group(order by t2.parentId) Ids
from ChildValues t1
left join ChildValues t2 on t1.sA=t2.sA and t1.sB=t2.sB and t1.ParentId<>t2.ParentId
where t1.Parentid=1
group by t1.ParentId