His parents are titled. The child's father is a Prince, he just chooses not to claim it.
In the BRF Princes are only the sons and male line grandsons of a monarch:
Charles, Prince of Wales
William of Wales
Harry of Wales
Andrew, Duke of York
Edward, Earl of Wessex
James of Wessex (not using the title at request of parents and with permission of the monarch)
Richard, Duke of Gloucester
Edward, Duke of Kent
Michael of Kent
Philip is an exception as he was created a Prince by the Queen in 1957 - 10 years after their marriage and 5 years after she became Queen but he had been born a Prince in his own right.
Princesses do not pass on titles to their husbands or their children thus Princess Elizabeth - the present Queen couldn't pass on a title to her children without new Letters Patent being issued just before Charles' birth - otherwise he would have been born Lord Charles Mountbatten, Earl of Merioneth (as a courtesy title - and he could still use that today as the heir to the Dukedom of Edinburgh) - through his father's title.
Margaret also couldn't pass on a title to her children so her husband was created Earl Snowdon so that her children would be born as Lord David Armstrong-Jones, Lord Linley as a courtesy as the heir to the Earldom and Lady Sarah Armstrong-Jones.
Anne and Mark chose not to have Mark given a title and so their children were born plain Master Peter Philips (now as an adult Mr Peter Philips) and Miss Zara Philips. They have no titles and are not entitled to any titles. As Anne says 'My children are not royal. They just have the Queen for a grandmother'.
In the next generation only William and Harry can pass on the title Prince/Princess (and in the current reign only William to his eldest son).
Beatrice, Eugenie and Louise (who like her brother doesn't use the HRH Princess title) hold that title because their descent from the monarch is male line - like the present Dukes of Gloucester and Kent, Prince Michael of Kent and Princess Alexandra - these four are male line grandchildren of George V. None of their children will be HRH Prince/Princess (unless something happens to William and Harry before they have a legitimate child in which case Beatrice would become the eventual heir).