There is no direct evidence in the text to suggest that Romeo has an opinion of how his parents will react to Juliet beyond the obvious issue of the feud, which he mentions frequently. From this we can infer that he does not expect them to be able to get over their long standing anger for long enough to form an unbiased opinion of her (interestingly, Juliet's father is able to see Romeo as a person independent of the feud and worthy of respect--we see this when he tells Tybalt to leave Romeo alone at the party, saying that Romeo has a good reputation in town and must not be fought with).
We know that Romeo does not talk to either of his parents about what is in his heart since when the play opens, both of them are concerned about his emotional state but don't know what has caused it. In fact, the only adult Romeo speaks to is the friar. From this we can infer that Romeo feels his parents would not understand, and would not approve of his relationship with Juliet.Of course he has not talked to either of them to determine where they would actually stand on the matter, and I think that is part of the point in asking this question. He can't possibly know how they'd truly react--he doesn't trust them enough to tell them.
Romeo probably expected to solicit some sympathy for his pining away for a girl he couldn't get. Certainly he would have liked for his parents to recognize the intrinsic qualities of Juliet for what they were - her beauty, natural innocence, etc...
In his heart, though, Romeo knows that his entanglement with Juliet will bring them nowhere, even though both of them strive to find a way out.
This is because of the family feud which has been going on between the Montague and Capulet families. There seems to have been some kind of 'Godfather' code of honour here forbidding anyone 'to let dead dogs lie.'
Curiously enough, in the end Romeo's father displays a true desire for reconciliation between the two families, demonstrated when he takes Capulet's extended hand and when he proposes to erect a gold statue in Juliet's honour. (see footnotes) In his character profile its seems logical to presume that Romeo would have been aware of this 'soft streak' in his father.
Rememer, though, that in fiction we are dealing with characters purely construed and not real people; we cannot know more that what the author pro-offers because this terrain is 'virtual' and only exists in terms of the reader's imagination.
o.m.g Romeo and juliet is the best play.