How to deal with parents who are noticeably partial towards some of their children,
but categorically deny it when informed regarding it. They may not know they commit such an act, but it is pretty evident. No matter what, the parents take sides with the younger children
against the eldest. This undermines the authority the eldest child has on any matter and loses respect in the eyes of their younger siblings. The parents have no problem admonishing the eldest in the presence of the younger siblings.
This regularly leads to the younger ones emboldened confrontations with the eldest, just because they are easily favored by the parents.
This causes the eldest one to become frustrated, exhibit immense anger and go silent. Even silence is criticized and considered "abnormal".
One of the parents picks and chooses what to follow in islam (mostly due to ignorance) and the other parent, despite being knowledgeable doesn't educate them when it is their duty to do so. When the eldest child informs them that something is wrong and is not from the sunnah, they immediately ask the child to follow fardh first as there are many other countless sunnah people do not follow. One of the parents has never showed any form of compassion to any of the children overtly but still seems to show affinity towards the younger children while on the other hand, maintains a written record of "abnormal" behaviors associated with the eldest child. The shortcomings of the younger children are not written down, likewise.
The parents regularly ask the eldest child to consult a psychiatrist as a form of mockery.
What is the way to deal with such parents in islam? And what does Allah and the Messenger (PBUH) have to say about parents who are partial to their children?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

First of all, parents are predominantly compassionate towards their children, keenly seek their best interests, and cannot tolerate any harm befalling any of their children, let alone that the parent himself harms them. If the parents act contrary to what we mentioned, then the child needs to take a pause for self-reflection and try to identify the cause of their mistreatment. Perhaps they misunderstood a certain behavior on his part and, in which case, he should clarify the matter to them, or he may have neglected their due rights over him, and so he needs to apologize to them, and there are other possible causes.

Secondly, you mentioned that your siblings are young and that you are the eldest. If this is the case, then it is normal that the parents show more compassion and mercy to the younger children, and perhaps even more love as well, due to their young age and weakness. If your parents display more care towards your younger siblings, this should not be a reason for you to harbor any grudge against them or feel frustrated because of it. Also, your parents should prevent your younger siblings from disrespecting you and should instruct and discipline them in this regard as needed.

Thirdly, there is no blame on either parent for carrying greater love for one of their children, because a person has no control over such inner feelings. However, the ruling is different with regards to outward actions. Muslim scholars mentioned that it is recommended for the parents to treat their children equally in all outward actions.

Fourthly, if a child notices some strictness and mistreatment on part of his parents, he should patiently endure it, keenly show dutifulness to them, and be kind to them. This is the due right of parents over their children in all situations because Allah, The Almighty, commands children to show dutifulness towards their parents even if they are disbelievers. He Says (what means): {But if they endeavor to make you associate with Me that of which you have no knowledge, do not obey them but accompany them in [this] world with appropriate kindness…} [Quran 31:15]

Al-Bukhaari allocated a chapter in his book Al-Adab Al-Mufrad entitled: "'Dutifulness to parents, even if they are unjust.' In this chapter, he cited a report on the authority of Ibn ‘Abbaas, may Allaah be pleased with him, wherein he said: 'There is no Muslim who has Muslim parents and is kind and dutiful to them seeking the reward of Allah, The Exalted, for that, except that Allah will open two gates (i.e. of Paradise) for him. If there is only one parent, one gate will be opened. If one of them is angry with him, Allah will not be pleased with him until that parent is pleased with him.' He was asked, 'Even if they wrong him?' He replied, 'Even if they wrong him.'" [End of Quote]

Fifthly, there is nothing wrong with the child giving his parents advice in a kind and gentle manner. If they accept his advice, then praise be to Allah; otherwise, he should let them be and continue to be good and kind to them.

Sixthly, a child should frequently supplicate Allah, The Exalted, to bless his parents with all that is good and to reconcile his relationship with them to be at its best. Verily, the hearts are in the Hand of Allah, The Almighty, and He turns them as He wills. We ask Allah, The Exalted, to help you earn the satisfaction of your parents and their good and kind treatment.

Allah Knows best.