May we get engaged ourselves without our parents' knowledge?

A promise to marry between two people is not forbidden, but hiding it from the wali and family is unwise and contrary to the purpose of khitbah.
In the Shafi'i school, a woman's marriage is not valid without a wali, so the family must be involved in the end anyway. Secret engagements often lead to unsupervised meetings, promises that cannot be kept, and conflict with the family. Involve the wali and parents from the start; if they object, listen to their reasons and ask a respected third party to mediate. If the wali refuses without a Shariah reason, there is a wali hakim procedure through the court. Never opt for a secret or cross-border marriage without registration. See counselling.