While large language models (LLMs) show promise in hospitality and tourism, their ability to provide unbiased service across demographic groups remains unclear. This paper explores gender and ethnic biases when LLMs are utilized as travel planning assistants. To investigate this issue, we apply machine learning techniques to analyze travel suggestions generated from three open-source LLMs. Our findings reveal that the performance of race and gender classifiers substantially exceeds random chance, indicating differences in how LLMs engage with varied subgroups. Specifically, outputs align with cultural expectations tied to certain races and genders. To minimize the effect of these stereotypes, we used a stop-word classification strategy, which decreased identifiable differences, with no disrespectful terms found. However, hallucinations related to African American and gender minority groups were noted. In conclusion, while LLMs can generate travel plans seemingly free from bias, it remains essential to verify the accuracy and appropriateness of their recommendations.