Is what I did correct, by continuing to wear the Esdal despite my parents' objection, and should I continue to do so even if they are displeased?

A Muslim woman is commanded to wear a covering hijab (definition). When she wears her hijab, she is performing an act of worship, obeying her Lord, and protecting herself. Since your parents are commanding you to wear an un-Islamic hijab (adorned and eye-catching), you are not obligated to obey them. If they are commanding you not to cover your face, then discuss it with them and clarify that it is safer to have complete covering when going out. If they insist, there is no objection to agreeing with them if you do not fear temptation from uncovering the face. You can carry a khimar with you and wear it when there are non-mahram (definition) men present or when fearing temptation.