Is it possible to have the House of Bourbon-Parma to rule over Sweden instead of Bernadotte, what would be the consequences of this in Europe?
The problem would come in that the Bourbon-Parma at least in theory would need to convert to the Reformed Church (can never remember if Sweden is Calvinist or Lutheran). This isn't like the princess of Denmark who married the prince of Parma and popped out the current queen of Romania (that princess was born to a Catholic mother, raised Catholic as per the marriage contract AIUI).
Maybe somehow Carola Vasa (or one of her aunts) marries a Bourbon-Parma prince (her father was against her converting in the first place), and somehow the Bourbon-Parma-Vasa heir marries Lovisa of Sweden (OTL Queen of Denmark) after Carl XV changes the succession laws to let his daughter follow him on the throne. It's a big handwave, but that's all I got.