Welcome to SE! I'm guessing you're fairly new to Civi and the comments so far, although accurate, may seem a bit cryptic. I'll expand a bit ...
The end result should be 15 contacts: 10 children, 5 parents and 10 parent/child relationships.
You said '... import the 10 children records first. I then did another import for the 5 parent record'. As per Pete's comment, you don't need to do 2 imports. Civi will import both the child and parent records at the same time using the 'child of' mapping shown in your screenshot.
External identifiers have to be unique across all records (not just those in your import). Your first import created child records with ext id's 1-10 and then the second one (first attempt) tried to create additional child records with ext id's 1-10 and failed because of the uniqueness requirement. Switching to 'Update' gets around that problem, but you can avoid it entirely by doing a single import.
You don't need external identifiers at all for this import and unless you need them for other purposes, I'd skip them entirely. Once you get this import done, if you try repeating the process for a different set of data but reusing ext id 1-10 you'll hit problems. The documentation includes an example for importing two parents, and that does need an ext id. (Rereading that, it would benefit from a simple example of importing relational data as well as the more complex one.)
For the parents, Civi will attempt to match those details to an existing contact. If it succeeds it will use that contact, otherwise it will create a new one. Civi determines what is a match using Dedupe Rules. You can find those at Contacts > Find and Merge Contacts. You can read about those here. The rule used for deciding if the parent details match is the 'Unsupervised rule', so you need to create a rule based on the data in your import for parents (ie some combination of first name, last name and phone). As Pradeep suggests, you can create a rule matching on first/last name and set that as the unsupervised rule.
If you do that and run the import you should find it creates 15 contacts with the right relationships. Confusingly, the import dialog lets you select a dedupe rule but this seems to be used only for the main contacts of the import (children), not the related contacts (parents) - hence the need to change the overall unsupervised rule.
Once you have done the import, go back and change your unsupervised rule to something narrower. In general, first/last name is not enough for uniqueness.
Alternatively, if you already have the data imported with duplicate parent records but don't want to reimport, you can run a manual dedupe using the 'First/Last' rule. See the link above for more info.
