Family of Shiva
Adi Parashakti herself married Shiva in the form of Sati and Parvati. Sati, a daughter of Daksha, was the first wife of Shiva. In popular legends, Daksha once organized a Yagya where he invited all gods except Shiva. In a rage, Sati cursed Daksha to be killed by Shiva and she jumped into the fire. Shiva was in grief, and he roamed around the world with Sati’s corpse which caused a cosmic imbalance. To restore the imbalance, Vishnu cut off the corpse of Sati to calm Shiva down. Sati's body parts fell on the earth and became sacred sites, now worshipped as the different forms of Shakti (Read: Shakti Peethas) along with different forms of Bhairava, incarnations of Shiva.
Sati reincarnated into Parvati as a daughter of Himavat, the king of the Himalaya Kingdom. Shiva doesn’t seek worldly desire and hence no action or reaction. Shakti, in the form of Parvati, makes him open his eyes and perform his cosmic duty. From regional or tribal deity to mainstream mother goddess, she is worshipped in various forms and manifestations together with different forms of Shiva or Bhairava, a fierce manifestation of Shiva. Shiva and Parvati are often presented with Ganesha and Kartikeya, their two sons. Padma Purana refers to three children, citing Ashokasundari as their daughter. Ashokasundari is a lesser know goddess who was created from Kalpavriksha, a wish-fulfilling tree, when Parvati wished for a daughter to overcome her loneliness. She is also associated with Bala Tripurasundari, the daughter of Tripurasundari (Read: Goddess in Sri Vidya)
Consorts of Kartikeya
Kartikeya is a warrior god and the first son of Shiva. He married Devasena and Valli, the reincarnations of Vishnu’s daughters Amirtavalli and Sundaravalli, respectively. Devasena, also called Teyvanai, is the daughter of Indra. Valli was a huntress born to Nambi Rajan, the chief of the Kuravar tribe, and his wife. Devasena is shown with a yellow complexion while Valli in green skin, linking her with ethnic dark skin. In many parts of India, Kartikeya is considered either unmarried or married to Devasena only.
Consorts of Ganesha
Ganesha is one of the best-known and most worshipped deities in the Hindu pantheon. Different texts give a different story about his birth. He was created by Parvati, according to Shiva Purana and Matsya Purana. In a legend, an elephant-headed goddess (or demoness) named Malini gave birth to him after drinking the bath water of Parvati.
Ganesha is associated with different consorts in different texts, usually portrayed as his Shakti. These women are more like the female personifications of his own attributes. Ganesha Purana and Mudgala Purana describe Siddhi (spiritual power) and Buddhi (intellect) to be his consort, while Matsya Purana mentions Riddhi (prosperity) and Buddhi. He is often shown with Siddhi and Riddhi. He is also associated with Ashtasiddhi, the eight spiritual attainments. The ashtasiddi are Anima, Mahima, Garima, Laghima, Prapti, Prakamya, Istva and Vastva. In Shakti tradition, they are addressed as eight goddesses who were summoned by Ganesha to attack demons.
Ganesha is shown in many iconography with distinct goddesses as his Shakti. In the list of thirty-two forms of Ganesha in Sritattvanidhi, six include Shakti sitting on his lap. These goddesses are shown without any distinctive personalities or iconography.