First calculation - realistic?

I don't understand it. The new rental situation is difficult with an additional €300/month but supposedly you have €1,000 per month left over??? But the financing works better then??

I understand it as meaning that the current new rental situation has become unbearable due to the new owner. Creating an even newer rental situation is difficult because of pets and children and would then be €300 more expensive. That together, current rent +300 + savings rate leads to the thought that one could also afford a loan installment.

I believe there is some truth to that, but I also think that staying might be the better solution.

Well, yes, but 300€ more rent would then be 1030€ including utilities. For Cologne/Bonn with 4 people and several dogs, that's quite okay and not expensive.

It will still take a lot of time (1.5-2 years) until the house is ready. Or longer, if the plot is missing and an existing property is not an option. Maybe move anyway, depending on how unbearable it really is at the moment.

I felt like recalculating again:

My parents have a net income of about €3400 together, but they only have a little over 10 years left until their well-deserved retirement.
They would contribute about €600 - 700
[..]
Whatever pension my parents receive, they should keep it and at most contribute a small additional amount (€200 - 300).

10 years €700, then 20 years €300. That adds up to €156,000 in total.

That is the amount available to them for their own living space. You would still have to deduct interest costs, etc. A semi-detached house as outlined in the original idea is of course completely unrealistic, of course. Maybe it would just be enough for a very small granny flat. But honestly, they cannot afford that and you don’t have any money left to cross-finance their living situation.