Technically a mortgage is not a credit to buy a house but a credit that is secured against a house (in case you default in your debt the creditor gets to sell the house to recover it). So there is nothing preventing you from getting a mortgage to pay some other costs (like some holidays, or a new car, or your next business)1.
Getting a mortgage to refinance your debts is nothing new. In fact it is rather usual, as people with debts may get a lower interest with a secured credit to cancel their high interest credits.
That said, your plan does not sound like a good idea:
If it works it will be ok, but it will not be much of an improvement in relation with renting a house. Think that for every point of increase in the cost of the credit (unsecured loans are more expensive than secured loans) you or your parents will be paying $1000/year (if a mortgage has a 3% rate and the line of credit has a 6% rate, that is $3000/year).
To work, everything must go as you expect. You must (and probably your wife) need to get jobs paying well enough, you can not have unexpected expenses2, the value of your home needs not to sink3... If anything fails, you do not get a mortgage and your parents are left paying the interest of an expensive unsecured credit.
You are putting yourself (or your parents) at a significant risk for no clear gain, so I would advise you to think about it a lot (like a year or so to see if you get a decent job)..
1Not that I say that those are a good idea.
2HINT: your wife has been on maternity leave.
3Usually banks do not want to give you a mortgage for the full value of the house but only for a significant fraction of it; let's say a 80% of the assessed value of the house.
That way, if you default they still can get all of its costs (not only the credit, but interest and the costs associated with getting your house) paid. If they gave you 100% of the credit, it could very well that the bank would not get enough money out of selling your house to recover all of their money.
Which means that if, since you buy the house until you get a mortgage the value of the market goes down, it may turn out that the bank does not offer you enough money to cancel your credit.