Some adblock-flavored blocklists only register domains as third party (e.g. EasyList).
i.e. ||domain.com^$third-party.
It seems that Pi-Hole ignores these, to allow their access as first party.
An option to extend the blocking to these would be nice.
As it stands, users have to manually filter the $third-party tag.
That won't happen: A filtering DNS server only sees DNS requests, i.e. it cannot be aware of any contextual information nor any other resources, like a browser typically retreives from a webserver handling the domain.
The easy list you linked contains many such elements beyond the domain list, e.g.:
A list containing "filters" or "options" ($ character followed by some rule) is not supported because Pi-hole can't do anything with this rules.
These filters/options target paths or elements inside the content, but Pi-hole only sees domains.
Since Pi-hole can't see the content, rules including a filter that depends on the content must be ignored.
These lists are intended to work with browser extensions that are able to see the whole page/content and remove or hide some elements.
Please note that you are requesting to automatically change entries containing $xxxxx into something else and use it anyway.
You are asking for Pi-hole to simply remove the filter part of an entry and use it, but removing that part will effectively change the rule into something completely different.
Example:
The rule ||domain.com^$third-party is asking to block the domain only if it is a "third-party" domain.
If Pi-hole removes the invalid part as you are asking, the rule would become ||domain.com^, which should block the domain unconditionally.
The 2 important points:
Pi-hole has no idea what a "first-party" or "third-party" domain is.
Pi-hole only sees one domain per request.
There is no page, no content and no context to verify whether a domain is a "third-party" or not.
That's why the whole line is considered invalid and ignored.
Even if Pi-hole were modified to accept this type of entry and alter them, these lists would function differently than they would if used in a browser extension.
They would over-block, causing issues for many users.
The list linked on your first post was created for browser extensions only.
Do not try to use it with Pi-hole.