This is kind of a tough question, because I'm not sure whether to address this as an anthropology question (with interesting animal behavior to look at) or a historical question. Since this is /r/askhistorians, I'm going to have to say first that the earliest historical records from Mesopotamia are about crops (and trade and all kinds of stuff). So, it makes sense from a historical perspective to say that farming is the oldest profession.

From an anthropological (prehistory) perspective, it's kind of difficult to tell exactly how much prostitution was going on. But in an interesting paper I saw ages ago (internet time, I guess), we can see chimpanzees exchanging meat for sex (and other things, like support in social conflicts). There was also an experiment (ctrl-f "something else happened during that chaotic scene", but the whole article is interesting) a while ago where a researcher taught some capuchin monkeys how to use a coin system and saw one of them exchange "something that looked like a coin" for sex "out of the corner of his eye", and it never happened again, so I'm not sure about that.

All of that is to say, that humans may have participated in prostitution before historical records existed. I would argue, though, that in order for prostitution to be...well, prostitution, that there would have to be an exchange of material goods (meat/berries/shiny rocks/whatever) or services (protection, I guess?), and therefore "professions" such as hunting or foraging would, logically, have to exist before prostitution became viable.

I read a book by Austrian sociologist and anthropologist Roland Girtler about prostitution. He argues that prostitution, if defined as a profession is not the oldest occupation. He does not automatically consider the exchange of sex for material advantages to be prostitution; e.g. it could happen within a marriage or between acquaintances. For Girtler the defining characteristic of professional prostitution is anonymity. Thus he concludes that prostitution requires the existence of cities with large enough populations so that not everybody knows everybody else. Or to put it in a somewhat breezy way: You won't have a brothel in a small village where there's always an old neighbor watching the people on the street.

In the end it's all a matter of definitions.