As the project Coordinator, help the engineer to know the number of men he needs to clear the area for planting grass.

Here's the solution to your problem:

Part a): As the assistant director of the hospital, we need to determine the actual land in acres for medical buildings and the maximum number of cars that can occupy the hospital parking at once.

Step 1: Calculate the land allocated for medical buildings. The total land is 243 acres. Land for medical buildings is 0.5 of the total land. $\text{Land for medical buildings} = 0.5 \times 243 \text{ acres} = \boxed{\text{121.5 acres}}$

Step 2: Calculate the remaining land and the land for planting grass. Remaining land after allocating for medical buildings: $\text{Remaining land} = 243 \text{ acres} - 121.5 \text{ acres} = 121.5 \text{ acres}$ Land for planting grass is one-fifth of the remaining land: $\text{Land for planting grass} = \frac{1}{5} \times 121.5 \text{ acres} = 24.3 \text{ acres}$

Step 3: Calculate the land allocated for staff quarters, waiting area, and parking area. This is the rest of the remaining land after allocating for grass planting. $\text{Land for SQ, WA, PA} = 121.5 \text{ acres} - 24.3 \text{ acres} = 97.2 \text{ acres}$

Step 4: Determine the land specifically for the parking area. The ratios are Staff quarters (SQ) : Waiting area (WA) = 3 : 2 and Waiting area (WA) : Parking area (PA) = 1 : 2. To combine these ratios, we make the 'Waiting area' part common. Multiplying the second ratio by 2 gives WA : PA = $2 : 4$. So, the combined ratio SQ : WA : PA = 3 : 2 : 4. The total number of ratio parts is $3 + 2 + 4 = 9$. The parking area corresponds to 4 parts out of 9. $\text{Parking area} = \frac{4}{9} \times 97.2 \text{ acres} = 4 \times 10.8 \text{ acres} = 43.2 \text{ acres}$

Step 5: Calculate the maximum number of cars that can occupy the parking area. One car occupies up to 0.02 acres. $\text{Maximum number of cars} = \frac{43.2 \text{ acres}}{0.02 \text{ acres/car}} = \boxed{\text{2160 cars}}$

Part b): As the project coordinator, we need to help the engineer determine the number of men needed to clear the area for grass planting in only one day. The land for planting grass is 24.3 acres (from Step 2 of Part a).

Step 1: Determine the work rate of one man. Given: 2 men can clear 0.5 acres in 2 days. This means 1 man can clear 0.5 acres in $2 \times 2 = 4$ days. $\text{Rate of 1 man} = \frac{0.5 \text{ acres}}{4 \text{ days}} = 0.125 \text{ acres/day}$

Step 2: Calculate the total man-days required to clear 24.3 acres. $\text{Total man-days} = \frac{24.3 \text{ acres}}{0.125 \text{ acres/day/man}} = 194.4 \text{ man-days}$

Step 3: Determine the number of men needed to clear the land in 1 day. To complete the work in 1 day, the number of men required is equal to the total man-days. Since we cannot have a fraction of a man, we must round up to ensure the entire area is cleared. $\text{Number of men needed} = \lceil 194.4 \rceil = \boxed{\text{195 men}}$