Jonathan Broom-Edwards of Britain leaped 2.10m to win gold, while Poland's Maciej Lepiato , Rio Games champion, clinched bronze.
The 18-year-old Kumar is competing in his debut Paralympics, set a new Asian record with a 2.07m jump.
This is India’s fourth medal in high jump after Nishad Kumar, Mariyappan Thangavelu and Sharad Kumar
T64 classification is for athletes with a leg amputation, who compete with prosthetics in a standing position.
T44, the disability classification that Kumar has but is eligible to compete in T64, is for athletes with a leg deficiency, leg length difference, impaired muscle power or impaired passive range of movement in the legs.
His impairment, which is congenital, affects the bones that connect his hip to his left leg.
The ongoing Games are turning out to be India's best ever and the nation has so far claimed two gold, six silver and three bronze medals.