Looking for mantras which are focused on Parvati like "Gaum Gaurimutrye Namah ".
I found a Mantra called Gauri Mantra, from the book Brihat TantrasAra. It's a Tantric Mantra. So, before initiation from the Guru one is not qualified to chant it.
From the above mentioned book's pp 491:
GaurimantrAh: Hrim gauri rudradayite yogeswari savarma phat | Dvithantah shodasArnohayam mantrah sadbhirudahritah ||
So, the actual Mantra is this - Hrim gauri rudradayite yogeswari hum swAhA |
If you count it's ShodashAksari .i.e a 16-lettered Mantra.
The Rishi, Chandah and DevatA for the Mantra is as mentioned below:
Shirasi parvatrishaye namah, mukhe gAyatrichandase namah, hridi shri gauryai devAtayai namah.
So, we get the Rishi as Parvat, the Chandah as GAyatri and the DevatA is of course Goddess Gauri.
For, Purashcharana, one is required to do 1 Lakh Japa of it, besides doing Homa etc as usual.
Asya purashcharanam lakshyajapah Ajyena dashAmsa homah ||
Note that it is a Tantric Mantra and as such it is meant to be chanted only after proper initiation.
If one chants the Mantra without initiation (DikshA), then not only it is fruitless as an act but it is also considered as an offence in the Tantras.
The GAyatri Mantra for Goddess Gauri/PArvati is this - Om mahAdevi cha vidmahe rudrapatni cha dhimahi tannoh gauri prachodAyath ||
Yet another GAyatri Mantra for the Goddess can be found in this Linga PurAna page and it is:
Om Ganaambikaayai vidmahey Karma saadhyaih cha dhimahi, Tanno Gauri Prachodayaat |
The Devi Rahasya Tantram's 2nd Patala gives another Mantra for Goddess Gauri and which is as follows:
The mantra of Gauri is is Om Shrim Hrim Glaum Gam Gauri Gim Svaha.
Since, this is also one of the secret Tantrik Mantras, the restriction regarding chanting of such Mantras which is already mentioned, applies to this one too.
Yes there are many mantras for worshipping her. Few of them are:
"Sarva Mangala Maangalye, Shive Sarvaartha Saadhike
Sharanye Tryambake Gaurii, Naaraayanii Namostute"
"Goddess Parvati is the auspiciousness of all that is auspicious. She is the consort of Lord Shiva, who grants every desire of one's heart. I adore such Devi Parvati, who loves all her children. I bow to the great mother, who has given refuge to me."
"Maata Cha Paarvati Devi, Pitaa Devo Maheshvara
Baandhavah Shiva Bhaktaacha, Svadesho Bhuvanatrayam"
"Goddess Parvati is the mother and the divine father is Shiva. The devotees are the children. The world is the creation of these divine beings and we live on the land of such celestial beings."