When Elizabeth II was born in 1926, her parents were already the Duke and Duchess of York. Her grandfather, King George V, had only recently ascended to the throne and her uncle, Edward VIII, was next in line for succession. However, when Edward VIII abdicated the throne in 1936 to marry Wallis Simpson - a woman who was divorced and therefore not eligible to be queen - Elizabeth's father became king instead, making her mother the new queen consort.
Interestingly enough, this meant that Queen Elizabeth II would one day be succeeded by her own son rather than her daughter - something that would have been impossible if her mother had been queen instead. In many ways, then, it was actually advantageous for Elizabeth's mother not to have been queen herself.