According to the International Food Information Council, allergy to tree nuts, which includes pistachios, is one of the most common food allergies among adults and children. It is important to work with a physician to ensure safe and proper diagnosis and management of food allergies, including allergies to tree nuts.

Here are summaries of some published science about pistachios and allergy:

A survey of people living in Iran in areas where pistachios were and were not grown found that those living in the growing region reported more allergies to pistachios, cashews and almonds. Twenty-one individuals with pistachio allergy were further tested. The authors conclude that pistachio-allergic people will very likely also react to cashews and almonds, and possibly other tree nuts.
Noorbakhsh R, Mortazavi SA, Sankian M, Shahidi F, Tehrani M, Jabbari Azad F, Behmanesh F, Varasteh A. 2011 Pistachio Allergy-Prevalence and In vitro Cross-Reactivity with Other Nuts. Allergol Int. May 25.

This report describes the allergic reaction of a 69-year-old woman to pistachios. She was also found to be allergic to cashew nuts and mango.
Ando K, Watanabe D, Tamada Y, Matsumoto Y. 2011. Oral allergy syndrome with severe anaphylaxis induced by pistachio. Int J Dermatol. May;50(5):632-3. (correspondence)

In a survey of 188 U.S. households with peanut or tree nut allergy, the number of subjects reporting allergy to each tree nut was: walnuts, 41; cashews, 29; pecans, 26; almonds, 25; pistachios, 19; Brazil nuts, 19; hazel nuts, 17; Macadamia nuts, 17; and pine nuts, 11.
Sicherer SH, Muñoz-Furlong A, Godbold JH, Sampson HA. US prevalence of self-reported peanut, tree nut, and sesame allergy: 11-year follow-up. 2010. J Allergy Clin Immunol. Jun;125(6):1322-6.

Four adults in Japan with allergy to cashew nuts were given skin-prick tests to determine if they were also allergic to pistachios, and found that there was cross-reactivity to pistachios as well. The authors note that cashew nuts, pistachios and mangos all belong to the same plant “family”, so may share the same proteins that can cause an allergic reaction in susceptible people.
Hasegawa M, Inomata N, Yamazaki H, Morita A, Kirino M, Ikezawa Z. 2009. Clinical features of four cases with cashew nut allergy and cross-reactivity between cashew nut and pistachio. Allergol Int. Jun;58(2):209-15.

This report describes how a 16-year-old boy showed signs of an allergic reaction to pistachios only after 30 or more minutes of exercise. Without exercising, all allergic reaction tests to pistachios were negative. The authors note that this is the first case of an exercise-induced reaction to pistachios.
Porcel S, Sánchez AB, Rodríguez E, Fletes C, Alvarado M, Jiménez S, Hernández J. 2006. Food-dependent exercise-induced anaphylaxis to pistachio. J Investig Allergol Clin Immunol.;16(1):71-3.