Yes. But, although you technically pay tax to both cities, Parma Hts give you a full credit for what you pay Cleveland, so you are not double taxed. But, since Parma Hts's tax rate is 3%, and Cleveland's is 2%, you will be paying Parma Hts 1% in addition to what you already paid Cleveland, thru withholding.
You normally only have to
file a return in the city you live in. It is not necessary to file one in the
city you work in. Your employer will remit withholding to the
city you work in, but you do not file a return unless you're claiming a refund or your employer didn't
withhold enough (both highly unusual). The city you live in will allow a
credit or partial credit for the withheld tax you paid to the work location
city. Parma Hts tax rate is 3%, but will only allow a credit of up to 2% (which just happens to be Cleveland's rate).
See http://parmaheightsoh.gov/en-US/Income-Tax.aspx
