Did Harry's parents want him to sacrifice himself?

Question

When Harry's parents, Sirius and Lupin appear to him through the Resurrection Stone they seem to know that Harry is walking to his death.

“You are nearly there,” said James. “Very close. We are . . . so proud of you.”

“Does it hurt?” The childish question had fallen from Harry’s lips before he could stop it.

“Dying? Not at all,” said Sirius. “Quicker and easier than falling asleep.”

“And he will want to be quick. He wants it over,” said Lupin.

Harry Potter and the Deathly Hallows

and then

“You’ll stay with me?”

“Until the very end,” said James.

“They won’t be able to see you?” asked Harry.

“We are part of you,” said Sirius. “Invisible to anyone else.”

Harry Potter and the Deathly Hallows

From the above I gather that Harry's parents, Sirius and Lupin all knew where Harry was going and were okay with the decision and supportive.

But I think that his parents wouldn't want their son to walk to his death even if they knew it was the only way to defeat Voldemort, much like Mrs Weasley wouldn't.

Are the people that come out of the Resurrection Stone figments of the imagination of the person who holds it or imprints of the real people? Does that dialogue above mean that Harry's parents had come to terms with the fact that he was was sacrificing himself which is a very unlikely thing even for a hero's parents? Or what they said to him was what Harry wished they'd said?

Answer 1

Yes

Harry's parents are already dead. They know that there is not much to fear from death. Thus they have much less reason than living parents not to want Harry to die. That said, they understand the impact that death has on those in the world of the living, and therefore wish for there to be an end to Voldemort's reign of terror. Being dead, they no longer fear death, but there are still other reasons to kill Voldemort. As Dumbledore (who is dead) says:

“Do not pity the dead, Harry. Pity the living, and above all, those who live without love. By returning, you may ensure that fewer souls are maimed, fewer families are torn apart.

—Harry Potter and the Deathly Hallows

That, presumably, is what Lily and James are proud of Harry for preventing.

The lines quoted in your question make their attitude clear:

“Very close. We are . . . so proud of you.”

—Harry Potter and the Deathly Hallows

They are proud of him because he is willing to sacrifice his life, which, even knowing of the existence of an afterlife, means giving up all his friends for an unknown fate. Don't forget that both Lily and James in some sense sacrificed themselves so that Harry could live, making it possible that they would have been proud of him even in life. In addition, given that they were dead, they might have looked forward to seeing Harry again.

Dumbledore's spirit had a similar attitude:

“Harry,” He spread his arms wide, and his hands were both whole and white and undamaged. “You wonderful boy. You brave, brave man. Let us walk.”

—Harry Potter and the Deathly Hallows

Answer 2

I think the second.

It's clearly stated that the so-called Resurrection stone cannot recall people form the dead.

“‘Yet she was sad and cold, separated from him as by a veil. Though she had returned to the mortal world, she did not truly belong there and suffered.
(Deathly Hallows, Chapter 21, The Tale of the Three Brothers)

Although it's not explicitly said "what" are the people coming from it, I think the power of the Resurrection stone is to give one's memories and love a physical form. But it's still his memories.

As "Sirius" says, they are parts of him. It's also said they are like Dementors, who are also, something like the essence of one's soul and happiness. For me, they are like projections of what's in Harry's heart:

“We are part of you,” said Sirius.
(Deathly Hallows, Chapter 34, The Forest again)

The dementors’ chill did not overcome him; he passed through it with his companions, and they acted like Patronuses to him.
(Deathly Hallows, Chapter 34, The Forest again)

Also, all they say are things that Harry himself knows.

All this makes me think your second option is the good one:

They say what Harry felt or wished they would have said.

(Also, seeing a lot of things in the books, I don't think it would be coherent to have Harry's parents and the men who were like true uncles to him being OK with him walking to his sacrifice, but I don't have enough evidence for this, it's more a general impression, so I won't develop it).