What is the point of fitting curves to the implied smile in the market? (Other than pricing exotics where the hedging instruments are vanillas). How does fitting a vol curve help you trade/market make vanillas? Isn't it rather tautological to use the market's implied vol as your theoretical value when trading?
Pricing of vanillas is basically interpolation of existing (or past) quotes. It is easier to interpolate in implied volatility space , than in price space. Reasons are we need to interpolate in multidimensional space (maturity, strike,forward, etc) and satisfy non-arbitrage conditions.
Using Black-scholes formula is convenient mapping which would also simplify satisfying the non-arbitrage conditions (positive density (convexity) , monotonicity vs strike).
In price space parametrisations would get more complex than in implied volatility terms (you can get good match in say some FX markets with just simple quadratic implied vol interpolation in log(K/F)/sqrt(T) strike (with some simple wings extrapolation).
Also implied vol parametrisations have advantage that parameters usually have easily understood effect to implied vol smile/skew.