It is most likely fake. Or rather, it's a list that has been circulating baselessly for years now. Or rather, it's a combination of two lists - one purporting to be from 1872, and one purporting to be from 1916, with some phrasing changed. (There are versions that put all of these in the 1916 (sometimes given as 1915) list - it's easy for the people passing it around to be loosey-goosey because of the fact that it doesn't appear to be based on anything.) It translates to:

1. During the term of your employment, you will not marry. (1872)

2. You will not consort with men. (1916.)

3. From 8pm to 6am you will be at home, unless you are attending a school function. (1916)

4. You will not loiter in ice cream parlors downtown. (1916)

5. You may not travel beyond the city limits unless you have permission of the school principal. (1916)

6. You will not ride with a man in a carriage or automobile, unless he is your father or brother. (1916)

7. You may not smoke cigarettes. (1916)

8. You will not wear bright-colored clothing. (1916)

9. You will not dye your hair under any circumstances. (1916)

10. You must wear at least two petticoats. (1916)

11. Your skirts must be no more than 5cm above the ankle. (1916)

12. You must maintain a clean and orderly classroom: sweep the floor once a day, scrub it with soapy water once a week, wash the blackboards once a day, start the fire at 7am so that it is warm by 8am. (Neither.)

You can find the original lists uncited in hundreds of unscholarly works about teaching/the history of education, and used in hundreds of local history museums as something that teachers in their village/town/county were held to. For instance, here the 1872 version is in an Arcadia book about Atchison County, Missouri; here is a version of the 1916 list in a German-American magazine. And in fact, both do sometimes turn up even in serious academic literature, where they are sourced from unreliable non-scholarly works. In English, the 1915/1916 list started to appear in the 1970s and the 1872 list in the 1950s. Typically, they're introduced vaguely: e.g. "These rules for teachers were posted by a New York City principal in 1872 ..." It's much more likely that these reflect what later generations thought was considered "respectable" in 1872 and 1916, rather than that they were actually enforced during those years. In this more modern context, an ice cream parlor is a place that is seen as completely innocent - even childish - and so presenting it as the equivalent of a nightclub or bar to the people of 1915/1916 makes it look as though they were hyper-sensitive to social danger, or as though the world was so safe that a harmless ice cream parlor was the most concerning place for a teacher to be seen.

That being said, there was some legitimate concern about ice cream parlors during the early twentieth century! As public places that offered little adult supervision late at night, they could be used for nefarious purposes. In the 1911 The Social Evil in Chicago: A Study of Existing Conditions with Recommendations from Chicago's vice commission, it was noted that "there is no doubt that conditions surrounding many of these candy and fruit stores and ice cream parlors in certain districts of the city are particularly dangerous to young boys and girls. In fact the court records show that a large number of young girls have been ruined in these places." Confectionery and ice cream stores in Chicago were supposed to bar boys under 21 and girls under 18 between 10pm and 7am unless they were with their parents, and there were supposed to be no screened off areas, presumably to prevent public ruining of young girls or actual prostitution. In one case, an investigator noted that "eleven girls and 9 boys were here at 11:20 PM. ... One of the girls flirted with the investigator and he sat down at the table with her. She said she was 17 years of age and hung out most of the time in (X1244) saloon. ... She is a prostitute and takes men to the (X1246) hotel, (X1247) street." In another ice cream parlor late at night, three teenage girls "were acting very giddy" and one of the boys with them essentially told the investigator that they were also prostitutes. This is only one subsection of the "Amusements" section of the "Child Protection" chapter of the report; other places of amusement that had the same issues were lake steamers, amusement parks, dance halls, and cheap theaters. Managers of theaters could pressure girls (and boys) to trade sexual favors for a chance on the stage; men outside could pull in children to watch a show and then take liberties in the dark; burlesque theaters allowed young boys to watch nearly-naked women dancing. In 51 cases of delinquency in girls examined in another section, four gave the cause/place of their "ruin" (violent rape, coercive rape, or consensual sex, we don't know) as nickel theaters, two in fruit stores, one in an ice cream parlor, and one in a dance hall. The ice cream parlor was not wholly the innocent public place we tend to see it as today, and it's possible that the list of rules for teachers does have some basis in truth.