You will need 2 configurations for this as the evaluator works per configuration.
Your first configuration should serialize the parent and exclude children. Your second configuration can include the child items and be set to use the `NewItemOnlyEvaluator:
For example:
<?xml version="1.0"?>
<configuration xmlns:patch="http://www.sitecore.net/xmlconfig/">
<sitecore>
<unicorn>
<configurations>
<configuration name="Example.ParentItems"
description="Serialize the parent item but none of the child items"
patch:after="configuration[@name='Foundation.Serialization']">
<targetDataStore physicalRootPath="$(sourceFolder)\$(layer)\$(module)\serialization\parentItems" />
<predicate>
<include name="ParentItem" database="master" path="/sitecore/Content/ParentItem">
<exclude path="ChildItem1" />
<exclude path="ChildItem2" />
</include>
</predicate>
</configuration>
<configuration name="Example.ChildItems"
description="Serialize the child items using the NewItemOnlyEvaluator"
patch:after="configuration[@name='Foundation.Serialization']">
<targetDataStore physicalRootPath="$(sourceFolder)\$(layer)\$(module)\serialization\newitemonly" />
<evaluator type="Unicorn.Evaluators.NewItemOnlyEvaluator, Unicorn" singleInstance="true"/>
<predicate>
<include name="Child Item 1" database="master" path="/sitecore/Content/ParentItem/ChildItem1"></include>
<include name="Child Item 2" database="master" path="/sitecore/Content/ParentItem/ChildItem2"></include>
</predicate>
</configuration>
</configurations>
</unicorn>
</sitecore>
</configuration>
Update based on your updated question:
If you want to be more specific about which child items are NewItemOnly
then you just need to be specific with your exclusions in the main configuration and also with your predicates in your NewItemOnlyEvaluator
configuration.