Unable to connect to remote service using AWS VPC Peering
How are you trying to connect to the remote service? - IP address, default internal/external DNS name or custom DNS name private/public hosted zone?
Please compare the output of nslookup to the remote service in both the VPCs. If the IPs resolving are different, the traffic is trying to go through the public internet and not the peering connection.
It would most probably be the need to either share the private hosted zone with the peered VPC or share internal hostnames across the peering connection