Since this blog does not believe in Chamatkaars(Miracles), all we can say is Satyawati was found on the river bank, along with lot of fishes caught by fisherman.
As said earlier, Parashar was highly knowledgably person. Omens and whatever information he had about Satyawati told her that this could be the mother of that child. He also realized that she wanted to get rid of her foul smell. He knew of herbal medicine which would do that. He gave those medicines to her and even directed her father how those medicines are to be administered Being an expert on Vedant Jyotish he could even visualize her later getting married to a king. He told her father his predicament. Her father was quite sympathetic to him and Satyawati too, was willing to help.
The second issue is neither the stature of Parashar, nor that of Satyawati father could have allowed for hiding the fact that Satyawati had conceived and was due to deliver a child. Forget the explanation given in some texts.