On the payment - I have a mom that pays monthly because she is charged a bank fee for every transaction so my fee is once at the beginning of the month instead of 4 weekly cheques. As long as it is in advance it is fine with me. My pay day is Monday and she counts up the mondays in the month - either 4 or 5 and pays accordingly.
Not sure mom really gets the deposit to hold the space issue. In her mind it makes sense but she is forgetting that she is holding you back from filling the space with someone else. If she isn't willing to hold the space with some sort of payment then she does risk losing it in theory.
Only you can decide if you can afford to go without the income for the four months or not or what the supply and demand in your area is. Back before it was common to get a holding fee I charged a family one weeks worth of care for each month I held the space which they had to come and pay me the first week of each month - allowed me to have contact with the family and often they came for a playdate the day she paid. Then once they started care the money was applied to the first week of each month till it was used up so if you hold for 4 months she would get the first week of the first 4 months as fees paid. So they got it back but not all at once and it was forfeited if they didn't stay in care for the full time.
I can understand why parents feel that it isn't their problem that you are losing money by giving them a space several months away and in reality it isn't. If we agree to hold then it is our responsibility to leave the space unpaid, fill it with a temporary child or ideally be filling a space for as soon as a child is leaving. What is normal in your area will determine which way you need to go unfortunately if you don't want to lose clients.