Connect a Plesk Lightsail Instance to a Mysql RDS Database

Check your route table has the correct IP route in your vpc/subnet route table back to the lightsail VPC. You will need to add one as this isn’t created automatically.

Without a route, the return traffic will follow the default route or one that has a better preference.

Is the Lightsail IP address set in the security group a private IP address?
Access via VPC peering, so if Lightsail's private IP address is not configured, you will not be able to connect.

Also, check again that you have a peer connection to the correct VPC where RDS is running.
It is also recommended to check the network ACLs used on the RDS subnet to be sure.