What is the way to convince the parents of a girl - who are not committed [to religious observance] - to agree to her marriage to a committed young man, knowing that they fear his commitment and wish to marry their daughter to someone who is financially "established," despite the girl's inclination towards the committed young man?

Mixing with the opposite gender, knowing its prohibition, is a grave evil. Following desires while being aware [of their wrongfulness] is extremely reprehensible. Allah has condemned those who take their desires as their god. Therefore, continuing your relationship with this girl and justifying it by claiming it's not a major sin is a serious matter. Persistence in minor sins transforms them into major ones, especially if it leads to infatuation (ishq), which is a debilitating disease. Prolonged lustful gazing and the ensuing infatuation can be more corrupting than a one-time act of fornication (zina) that is not persisted in. Infatuation is a weakness in the love of Allah and a weakness in faith. All your justifications about cooperating in prayer and remembrance are from the stratagems of Satan. You must immediately repent to Allah and show gratitude for the blessing of memorizing Allah's Book by appreciating it and staying away from suspicious and forbidden situations, for the status of a حامل القرآن (memorizer of the Quran) is great. The solution is to propose to the girl and expedite the marriage, or completely sever the relationship if a delay is requested, and seek someone else.