Hello Paris -
Apparently TurboTax chose to only have NYS E-File extension available inside the program.
However you can go on your browser to
https://www.tax.ny.gov/pdf/current_forms/it/it370_fill_in.pdf
& download the form, fill it in using Acrobat Reader, and then print it and mail it in. That is what I am doing. Instructions (like where to mail the form) are on Page 3 of the form. You should print pages 1-2 back to back if you can. You can use 'Save As' under "File" in Acrobat Reader to save your filled-in form, if you want too open it later & reprint it for instance. According to the instructions, this is totally acceptable as an extension request & you can then E-File your return via Turbo Tax later.
You can file an extension to file for New York in TurboTax CD/Download. If you use TurboTax Online, you can use this link to file one.
More information is available in this TurboTax Help topic .