It remains unclear what exactly happened back in time.
The Volkskrant:
"Carlos en Gitte zijn weliswaar jeugdvrienden, maar ze zijn niet getrouwd en wonen niet samen. De baby blijkt een wens van Gitte. De geboorte wordt stilgehouden, maar lekt na bijna twee maanden uit via De Telegraaf. Carlos laat in een verklaring weten dat het 'een eigen, zelfstandige beslissing van Brigitte was om moeder te worden', een beslissing die hij zei 'volledig' te respecteren. Hij betwist niet dat hij de vader is, maar erkent het kind niet. Hij verklaarde ook dat het 'uitgesloten' was dat er 'een familierechtelijke betrekking' tussen hem en het kind zou komen."
Translation:
"Carlos and Gitte are indeed childhood friends, but they are not married and do not live together. The baby turns out to be a wish of Gitte. The birth is kept private, but after almost two months the news leaked via De Telegraaf. Carlos states in a statement that it was 'Brigitte's own independent decision to become a mother', a decision he said to 'fully' respect. He does not dispute that he is the father, but does not recognize the child. He also stated that it was 'excluded' that there would be 'a family relationship' between him and the child."
And De Volkskrant again:
"Op verzoek van de moeder legt de rechtbank het vaderschap na twee jaar vast. Gitte wil niet dat de vader zich nog met de opvoeding bemoeit, waarover Carlos later zegt dit te hebben betreurd. Hij draagt wel financieel bij."
Translation:
"At the request of the mother, the Court of Justice establishes paternity after two years. Gitte does not want the father to interfere with the upbringing, which Carlos later says he regretted. He does contribute financially."
In 2015 the Minister of Justice agreed with the request of Carlos Klynstra to be known with his father's surnames and title.
De Volkskrant again:
"Carlos dient daarop een bezwaarschrift in. Aan The Post Online laat hij eind 2015 weten: 'Ik heb vernomen dat Hugo Klynstra een dergelijk verzoek heeft ingediend. Dat verzoek is op mijn initiatief nu onderwerp van een procedure. Het gaat om een persoonlijke aangelegenheid met een voor mij verdrietige achtergrond."
Translation:
"Carlos submits a notice of objection. At The Post Online, he announced at the end of 2015: "I have been informed that Hugo Klynstra has submitted such a request. That request is now the subject of a procedure on my initiative. It is a personal affair with a -for me- sad background. "
De Volkskrant again:
"Carlos' dynastieke bezwaar dat de naamswijziging niet past binnen de regels van het Huis Bourbon-Parma, vindt de rechter een particuliere aangelegenheid die niet onder Nederlands recht valt."
Translation:
"Carlos' dynastic objection that the name change does not fit within the rules of the House of Bourbon-Parma, is seen by the Judge as a particulier case without jurisdiction under Dutch law".
Brigitte ("Gitte") Klynstra, a stepdaughter of the Count van Rechteren-Limpurg, died in 2010 at the age of 51. I am intrugued by the sentences "the baby turns out to be a wish of Gitte" and "it was Brigitte's own independent decision to become a mother". The two agreed not to have any forrmal relationship, Brigitte did not want any involvement of the father with Hugo (but Carlos had to pay anyway). The Duke regretted that he was excluded from his son. All by all a sad history with dynastic complications, as the Duke himself of course is the head of the House and applied the rules as he would have done to another dynasts with extra-marital children.
It is for 99% sure that Hugo will receive the title and surname prins de Bourbon de Parme which is part of the Dutch nobility and on which the Dutch legislation concerning names and titulature applies. Most likely the Duke only wanted to make clear he has done his "duty" as Head of the House , that he has done "his dynastic obligations", while knowing on beforehand that Hugo will "win" the case anyway.
Those are beautiful photos of the family, thank you for sharing iceflower. It's interesting because the children all resemble one of their AAA cousins: Luisa resembles Princess Alexia, Cecilia Princess Ariane and Carlos Princess Amalia. There are some very dominant genes in the DRF. 
