First of all, the D/D method is called D/D because it is "diffusing and defusing". Diffusing in the sense of universalizing, spreading out, and understanding the universality of the human condition, the universality of Dukkha. Defusing in the sense that it takes the fuse out of the emotional bomb that we hold inside, by understanding that it's not just me who is subject to these things, but all beings. That aging, disease and death and separation from loved ones is universal, and that it's not just my father but it will be me some day. It's not just my father but everybody's father eventually. So this is the D/D method, which helps.
When is it appropriate to share the Dhamma? This is a very fine balancing act, it's very difficult to know when it is appropriate. It depends on the situation, on the people involved, on how open you are, how close you are to them and how open they are. I believe it's quite important first to sympathize with the situation without giving our opinions, so that the person knows that we are feeling with them at such a difficult time, and we are sympathizing with this very difficult condition. That way they understand that we're sensitive to the situation and we're not just trying to give advice which they may not wish to hear. So we have to be sensitive enough to know whether they are wanting our help, or whether they are not wanting our help. Yet it's often better to say something than nothing, even in this difficult situation of not knowing when to say something and when not to say something. Many people get silent because they don't know what to say, and that's often the worst approach. Frequently this is because they are only thinking of their own uncomfortable feelings rather than thinking about the difficulty of the other person and the situation.
I knew a woman whose husband died, and then her friends stopped coming to see her, because they didn't know what to say and it made them feel uncomfortable.
That's the worst situation, because then this woman didn't feel the support that was necessary at that time. So even in our confusion about whether to help or not to help, I believe it's better to actually show that we care by talking and being with them, and being sensitive to what they are saying. Many times in situations like this there is the tendency to pull away from the Dukkha and the uncomfortable feelings that we have, and not wanting to deal with them.
So I believe it would be helpful to say something to get them talking, so that you can see what they are thinking. Then, if you see what they are thinking you may be able to put in an appropriate story to diffuse it, rather than saying "this is the D/D method and you can use it". To actually give a story at that time is very helpful. This is actually how the Buddha approached such a situation, if you know the story of Kisagotami in the scriptures.
For those who don't know the story, Kisagotami lost her only child and went into deep grief and shock. She went to the Buddha, and the Buddha asked her to go and get a mustard seed so he could heal her affliction. Before she left, the Buddha said the mustard seed must come from a house where no one has died. Kisagotami didn't understand significance of what the Buddha was saying at that time, but she went to all the houses in the village and asked for this mustard seed. Yet upon asking the question, she always got the answer, "I'm sorry, my father died last week," or "I'm sorry, my grandfather died" or "I'm sorry, my brother died.", etc. She went to every house in the village, and she couldn't find the mustard seed where no one had died. Through this, she came to understand the universality of death, and returned to the Buddha. The Buddha asked her, "Kisagotami, did you find the mustard seed?" She said, "I could not find a mustard seed from any house where no one had died". The Buddha then gave her a teaching on impermanence, and apparently she reached the first stage of enlightenment.
However I would think it is better for us not to expect that everybody reaches enlightenment if we give them the D/D method! Yet to give them a little story may be helpful so that they can understand the universality of Dukkha, it happens to us all. So I believe it is better to say something rather than nothing, to be sensitive to the situation, so that you know who you are talking to, where they are coming from, and how open they are. So there is not a "yes or no" answer to this question. Which is often the case of many of these situations.