Do humans have skin cells that trigger oxytocin release in the brain?
Short answer
Oxytocin release has been associated with the cutaneous low-threshold (CT) afferent fibers in hairy skin.  

Background
Oxytocin is released from the paraventricular nucleus in the brain in response to low intensity stimulation of the skin, such as touch, stroking and warm temperatures. 

In a recent review Uvnäs-Moberg et al. (2015) report that the specific sensory nerve type mediating oxytocin release in the brain is unknown. Although myelinated somatosensory fibers mediating the discriminative sense of touch may be involved, a more likely candidate mediating affective touch are the unmyelinated low-threshold mechanosensory nerves, called C low-threshold mechanoreceptors (CLTMs) in animals or C-tactile afferents (CTs) in humans. Their terminal morphology (i.e., the skin receptor they connect to) is currently unknown. The fibers have been characterized mainly through  electrophysiological and electron microscopy research (McGlone at al., 2014).  

The association of CT fibers with affective touch stems mainly from the finding that their activation patterns with various stroking velocities correlates with subjective pleasantness ratings. Specifically, the relationship between brush-stroking velocity and firing rate was distinctly different between CT and myelinated afferents. CTs showed an inverted U-shaped relationship,
whereas mean firing rates increased monotonically with brushing velocity in all myelinated afferents (McGlone at al., 2014).

However, CT fibers have only been identified in hairy skin in mammals, and the fibers mediating affective touch through glabrous skin (including the fingertips, palms and lips) are unknown (McGlone at al., 2014).  

References
- McGlone et al., Neuron (2014); 82: 737-55
- Uvnäs-Moberg et al., Front Neurosci (2015); 5(1529): 1-16