We propose a quasi maximum likelihood estimation method for Bergomi-type stochastic volatility models with parametrized kernels, focusing on the estimation of the kernel parameters from high-frequency time-series observations of option prices. We first show that the cumulative forward variance, which can be reconstructed from option prices, solves an infinite-dimensional stochastic differential equation driven by a one-dimensional Brownian motion under the Bergomi-type model. To overcome this degeneracy, we introduce a nondegenerate proxy likelihood based on the Euler-Maruyama approximation and define an estimator through the associated estimating function. We establish consistency and asymptotic mixed normality of the proposed estimator under a regular class of kernels. Simulation studies and an empirical application to SPXW option data illustrate the finite-sample performance of the method and the practical relevance of the approach.