We measure nonlocal third-order spatial correlations in nondegenerate ultracold gases of bosonic (⁸⁴Sr) and spin-polarized fermionic (⁸⁷Sr) strontium through studies of the formation rates for ultralong-range trimer Rydberg molecules. The trimer production rate is observed to be very sensitive to the effects of quantum statistics with a strong enhancement of up to a factor of 6 (3!), in the case of bosonic ⁸⁴Sr due to bunching and a marked reduction for spin-polarized fermionic ⁸⁷Sr due to antibunching. The experimental results are compared to theoretical predictions and good agreement is observed. The present approach opens the way to in situ studies of higher-order nonlocal spatial correlations in a wide array of ultracold atomic-gas systems.