Could my parents change my birth name without me knowing?
This generally requires a court order (everything depends on jurisdiction: this is a state matter, not a federal matter). As a minor, the courts could allow your parents to change from Dweezil to William without involving you, until you are old enough that the judge thinks you might be able to have reasonable input into the matter. Once you're over 18, your parents can't change your name – you would have to do that, at least if you are mentally competent.
In Washington, the courts juggle the wishes of the child, the wishes of the parents, how long the child has had the name, and the social advantages or disadvantages of the name change, and permission from the minor is required if over 14 (child input would be solicited for a child over 7). Since this involves a court order, in principle this information is available to the child. In cases involving domestic violence, the records could be sealed.
A name can be changed by changing the birth certificate which means filling out a form and paying a fee, and if the child is under 1 year old, it just requires the signatures of the parents (or, a court order). This "under 1" paperwork approach seems to be widespread (Colorado, New York, others). Also bear in mind that the initial filing of a birth certificate may well not have a child's name, which may not be supplied until the parents make up their minds.
Changes to the birth certificate are knowable (they don't erase anything), but can only be revealed to the subject of the record, or in case of court order. Thus a change should be discernible, if other states are like Washington.