This can't exactly be considered a definitive answer, but here's what I could find. It's a bit of a roundabout story.
The original Shiva Purana discusses the marriage and courtship of Shiva and Parvati in Section 2.3, which is the Parvati Khanda of the Rudra-saṃhitā. There are only two references to dates in the whole section. The first comes in Chapter 39, line 16, where Shiva tells Narada:
O Nārada, the marriage will take place after seven days from today. Following the worldly conventions I shall make a grand festival of the same.
Unfortunately, no mention of the current date then is given in that passage, so it turns out not to be so useful.
The other mention is in Chapter 48, which describes the ceremony itself.
4] Then the brahmins were requested by Himavat:
“May the rite be formally started after narrating the Tithi etc. The auspicious hour has come.”
5] After saying “So be it”, the excellent brahmins who knew the proper time proclaimed the Tithi etc. very delightedly.
There, the tithi is said but not specified. This is the extent of the information in the Shiva Purana.
(The other text that devotes itself to Shiva and Parvati's marriage is the Kumarasambhava by Kalidas, but this is also lacking in the specifics of the tithi.)
Then I found this book about the Agni Purana, where the author writes:
The Purana reforms that it was on the third day of the lunar fortnight, i.e. 'triteeya' that Lord Shiv married Parvati or Gauri. Rites performed on this day are thus known as 'Gauri Vratha'. Lord Shiv and Gauri have to be given only of the offering of front. On this day eight names, given below, of the Gauri ought to be ritually recited. The names are: Lalita, Viyaya, Bhadra, Bhavani, Kumnda, Shiva, Vasudeni and Gauri.
And in a footnote:
This information is in the sharp contradiction of the normal schedule. Normally Lord Shiv's marriage day is solemnised on Shivratri day i.e. 14th day of the bright fortnight.
It took a long time to find the actual chapter, but I eventually found it. It's in the Agni Purana's section about vows to be made on certain days; this one is in the section for vows to be made on the third lunar day:
1] I shall describe the vows (to be practised) on the third lunar day which would confer enjoyment and emancipation. Listen to me. (I shall describe) the mūlagaurīvrata for the (goddess) Lalitā (a form of consort of Śiva) (to be done) on the third lunar day.
2] Goddess Gaurī was married by Hara (Śiva) on the third day of the bright (fortnight) in (the month of) caitra (April-May).
It's a little confusing because it says 'marriage by,' but this seems to have been an old variant of 'married to.' The original Sanskrit chapter also says this.
So the answer seems to be the third day of the Chaitra month.
A few random notes:
- This version of the Shiva Purana says that the chosen day was the thirteenth day, or trayodashi of Phalguna. But there's no reference in the actual Purana to support this.
- This book on Kashmiri culture says that they, too, celebrate it on trayodashi. Wikipedia says that this is a feature of Kashmiri Shaivism.
- The Wikipedia article for Maha Shivaratri, linked above, says that there are a lot of different thoughts about Maha Shivaratri's significance, and one of them is that it marks the anniversary of Shiva and Parvati's marriage. Many other online sources agree.
There may be more to it, or there may not, but that's the best I can do.