If I rent a property from someone and sublet it to a third party for rent because I could not or could only partially occupy it. Is the rent from subletting considered house property income?

No. Such rental income cannot be considered House Property Income.

Only when you are the property owner, you can claim the rent as "Income from House Property" for tax purposes.

Since you are the tenant in the aforementioned situation and not the property's owner, the rent that you earn from subletting the property whole or in part will not be taxed as "Income from House Property" but rather as "Income from other sources."