Three Possible Questions
I'm not entirely clear if you are asking (1) which courts the Hague Convention is enforced in (which I answer below), or (2) if you are trying to distinguish between (a) enforcement of parental decision making responsibility, i.e. on issues like permission to marry and major non-emergency medical decision-making, and (b) enforcement of physical parenting time, i.e. where the kid is supposed to be physically located pursuant to a court order related to custody of a child, for example, as a result of a divorce or in the case of an unmarried couple who are parents of a child (which I do not answer).
I have retained the answers to all three questions in this answer, even though you seem to be primarily concerned about question (2).
The Answer To Question (1)
The Hague Convention you identify is generally enforced in the domestic courts of the country where the child is physically located that has child custody jurisdiction or via a "Central Authority" as defined by a signatory to the Convention, although it is a bit more complex than that because the Hague Convention has one set of rules for "emergencies" and another for ordinary cases.
In "emergencies" the relevant court in the country where the child is actually present can act without regard to other court orders that are in force. In non-emergency cases, a court where the child is located is supposed to follow court orders made by the court with jurisdiction over the "home" of the child as defined in the Convention, even if that means enforcing a foreign court order rather than making an independent determination regarding custody of the child.
In a case where a child is physically present in the United States, the relevant domestic court would typically be a state court of general jurisdiction or a "family court" of that state, depending upon the U.S. state. In another country that is a signatory of the convention, it would be the corresponding court of that country.
There is a fair amount of case law interpreting the Hague Convention in multiple countries (including multiple U.S. Supreme Court cases), but it is not easy to summarize and I don't know of a free Internet accessible resource that organizes and summarizes that case law in one easy to find place the way that an annotated statute set, or international domestic relations textbook (assuming the any exist), or legal digest would.
The Answer Question (2)
Article 8-20 are about "custody" while Article 21 is about "rights of access."
These terms are defined in Article 5 which basically says that "rights of access" mean what is commonly called "visitation":
a) "rights of custody" shall include rights relating to the care of
the person of the child and, in particular, the right to determine the
child's place of residence;
b) "rights of access" shall include the right to take a child for a
limited period of time to a place other than the child's habitual
Quebec is saying in its interpretation that if it isn't physical visitation, don't use Article 21. Parental authority is a "right of custody" as defined in Article 5 and also in the Quebec interpretation which fits the sense of the language in the Convention.
Of course, this is basically just a rules of the road interpretation. It isn't that you don't have a remedy under the Convention for a "custody" issue as that is defined.
It just means that you used Article 8-20 and not Article 21 to deal with "custody" as opposed to visitation issues, presumably with a less expedited process than for Article 21 cases.
But, you wouldn't be wrong in thinking that Article 8-20 are really tailored to physical custody and don't address how questions of parental authority as you use the term are enforced since there isn't a wrongful removal or retention, which is what those Articles seem to address. There is an obvious analogy to follow in a parental authority case, but the enforcement procedure is not made clear by the Convention (or the Quebec interpretation of the Convention), except insofar as it is not the remedy under Article 21.
By default you may be left resorting to Article 29 in these cases, i.e. non-Convention remedies in domestic courts of the country where the child's "home" is located, rather than the Hague Convention process where there is not a wrongful removal, there is not a wrongful retention, and there is not a physical visitation issue. Article 29 states:
This Convention shall not preclude any person, institution or body who
claims that there has been a breach of custody or access rights within
the meaning of Article 3 or 21 from applying directly to the judicial
or administrative authorities of a Contracting State, whether or not
under the provisions of this Convention.
Arguably, Article 29 has an affirmative effect by serving as a de facto full faith and credit clause for foreign domestic relations orders made by a court that had jurisdiction within the meaning of the Convention, rather that merely stating something that would be true even if it hadn't been said, and rather than merely not divesting the courts of a signatory of an alternative basis of jurisdiction that would have existed even if the Convention had been been signed.
After all, a breach of parental authority rights is still a breach of custody rights as defined in the Convention, and Article 29 affirmatively states that there is a right to enforce them via normal domestic court channels "whether or not" the enforcement rights arises under the Convention, which means that rights arising under the Convention can be enforced in that manner.
Alternately, a judge could consider maintaining physical custody in a manner other than that found in a binding custody order to be a wrongful retention, but honestly that stretches the language of the Convention pretty far from is plain meaning.
The Answer To Question (3)
It also occurs to me that you may be asking (3) about whether publicly funded attorneys or officials will help you enforce your rights under the Convention in the name of a right to "request assistance". But, subject to the discretionary generosity of your country's diplomatic corps when your child is in another country, generally, "request assistance" in the sense of the Convention means the ability to request that a domestic court in a country other than your own issue court to enforce your rights through litigation conducted at your own personal expense.
There is not a "civil Gideon" aspect to the Convention, although there are provisions address the issue of legal fees, fee shifting, and a foreign parent's access to generally available civil legal assistance in a foreign country.