Persian Is a Cousin, Not a Parent
Urdu did not come from Persian, and Hindi is not Sanskrit alone. The family tree tells a better story.
A common story goes like this: Hindi comes from Sanskrit, and Urdu comes from Persian and Arabic. The family tree says otherwise.
Linguists sort languages by ancestry. Persian sits on the Iranian branch of the Indo-European family. Arabic is not Indo-European at all; it is a Semitic language, from an entirely different family. And Hindi-Urdu sits on the Indo-Aryan branch. It descends from the old speech of the midlands, through Prakrit and Apabhramsha, to the Delhi vernacular called Khari Boli, the shared base of both modern standards.
So Persian is not the parent of Urdu. It is a cousin of the whole family, and a close one. Indo-Aryan and Iranian drifted apart roughly four thousand years ago, and their oldest scriptures still rhyme. Scholars have shown that verses of the Avestan hymns of Iran can be turned into intelligible Vedic Sanskrit by regular sound substitutions alone.
What Persian handed Hindi-Urdu is a wardrobe, not a bloodline, and the language has worn it so long it would not be itself without it: a wealth of loanwords, many of them Arabic words that traveled to Delhi dressed in Persian, borrowed sounds like z, f, and the three guests who stayed (see Kh, Gh, and Qaaf), the Perso-Arabic script that Urdu writes in its flowing Nastaliq hand, and whole genres of poetry. English did the same with French and stayed Germanic; by one classic count over half its dictionary is French and Latin, while nearly all of its hundred commonest words stay Anglo-Saxon.
Ask a pronoun, a number, or the verb hona where they were born. In Hindi-Urdu the answer is the same: here.