When a continuation patent is approved, will the parent patent still hold the rights to sue others if a product infringe the parent application and not the continuation patent
They are two separate patents, each with their own claims. As an example of their independence, the owner can, unless a terminal disclaimer was signed, sell the patents to two different entities.
The claims must be patently distinct but one patent might have claims to a method of making something and another patent might be to the thing itself. Also two patents might have claims that attempt to cover essentially the same thing but with a different approach to defining the thing.
If the claims are too close by one criteria (its complicated) they may be declared double patenting under a judicially invented doctrine. In that case both patents can be issued as long as the owner promises to always keep them commonly owned and co-terminus by signing a terminal disclaimer. If they are too close in a different way it is double patenting by statue and the second one would not be allowed.