Shakespeare, like other artists and playwrights of his time, was in a very tricky situation! When writing, authors had to be very careful of putting down their views - much more careful than today's politically correct times for example. You would be more likely to be hanged or beheaded than get a fine for upsetting the monarch back in those times. So playwrights were used to couching their opinions carefully. We can only look at the examples he gives - for example in Othello also, the father is patriarchal and dogmatic. Shakes peare was just 'telling it like it was' for women in those times - daughters like Desdemona and Juliet were expected to be obedient, subservient, meek, mild and sacrifice their own needs for the good of the family name and fortune. He does, however, show the anguish and emotional pain these girls suffered, which is good for us to see - and suggests he felt some empathy with them.
This is hard to really say, because it is not always apparent as to which things the characters do are things that Shakespeare himself agrees with and which things are not.
I want to focus here on the Capulets -- Juliet's parents. Early in the play, when things are still good and everyone seems pretty happy, the Capulets are not too demanding. They are willing to let their daughter postpone marriage if she wants and are, in general, seemingly interested in what she thinks.
Later in the play they are really quite abusive when she does not want to marry Paris right away. Lord Capulet, especially, is very violent in his language.
So which did Shakespeare approve of? I think he approved of the earlier attitude, putting it, as he did, farther from the tragic ending.
I suppose you can see a similar thing going on when he does not let Egeus get his way in "A Midsummer Night's Dream." So perhaps you can argue based on these two plays that Shakespeare thought parents should take their children's wishes into account.
Shakespeare demonstrates a nuanced attitude toward parenting, clearly illustrating the prevalent attitude of the time (children are their parents' possessions and owe them absolute obedience) as well as subtly allowing us to empathize with the frustration of the young people. We lets us into the tender protectiveness of 'good' parents as we watch the Montagues solicit Benvolio's help in getting Romeo to share his teenage angst, and even more so when we sit in on Lord Capulet cautioning Paris against rushing the very young Juliet into the same sort of tense situation Capulet has with his Lady. Lady Capulet, no one's idea of an earth mother, surely has Juliet's best interests in mind as she encourages her pliant daughter to 'marry well.' The Nurse, who is far more truly maternal, understands her young charge perfectly well and allows herself to get swept away by the passion of the romance, yet even when she 'betrays' Juliet by advising her to commit bigamy, she's motivated by wanting what's best for her. All this has to be weighed against the shockingly abusive Act III Scene V, and the terrible mourning of both sets of parents at the tragic ending is clearly unfeigned.
Egeus is a much less nuanced figure. Whatever elements he might have of empathy are never revealed by Shakespeare, who leaves him at being an overbearing tyrant. But MND is a play whose characters are almost entirely ciphers, deliberately so. The only ones that have any dimension of character are the supernatural ones.
As always, Shakespeare is a master at encouraging his audience to see his characters are real people, which is why they have endured for so many centuries.