What is the basis for these plants needing to 'stay away' from each other?

? I don't know . I was asking you . Basil and cucumbers are enemy's on companion charts I look up .

The reason I was looking to plant parsley with cucumbers was for a ground cover . I wanted to trellis the cucumbers and have an edible understory. I'm open for suggestions.

I meant what did the sources you have read give as the reason? Usually such sources don't give any solid evidence for their assertions that a plant 'likes' another one. Personally, unless a reason is given, I am very sceptical about 'Companion Planting.'

If two plants need similar soil, watering, light or such, then it makes sense to plant them together. But there is no reason not to plant 'strong herbs' together. They will not affect each other. Parsley is biennial and cucumbers are annual. So, it is possible that removing the cucumbers at the end of the season could disturb the parsley. But I can think of no other reason not to plant them together providing they have sufficient space.