If you are going to design table schema then you should use normalization
to optimize.
There are no arrays in MySQL and also there is no need for it.
First off all, if a "column in a RDBMS is meant to be atomic", in that it contains one and only one piece of information. Trying to store more than one piece of data in a column is a "violation of first normal form".
So you should have to alter your logic like below,
-------------------
|Cate_id |Parent_ID|
--------------------
|1 | 0 |
--------------------
|3 | 1 |
--------------------
|2 | 0 |
--------------------
|4 | 1 |
--------------------
--------------------------
|Product_id | Category_id|
--------------------------
|1 | 1 |
--------------------------
|1 | 3 |
--------------------------
|2 | 4 |
--------------------------
Hope this query will solve your problem then.
Select prdct.Product_id
from Products as prdct, Category as ctgory
where
ctgory.Cate_id = prdct.Category_id
and prdct.category_id =1
and ctgory.Parent_ID=1;