Does GridSearchCV not save the best parameters?
It doesn't, please try the following code
CELL1:
import numpy as np
from sklearn import svm, datasets
from sklearn.model_selection import GridSearchCV
iris = datasets.load_iris()

CELL2:
%%time
parameters = {'kernel':('linear', 'rbf'), 'C':np.linspace(0.1,100,1000)}
svc = svm.SVC()
clf = GridSearchCV(svc, parameters)
clf.fit(iris.data, iris.target)

CELL3:
%%time
clf.best_params_

Wall time of CELL2 will be about 7-9 seconds.
Wall time of CELL3 will be 0ns. ( instantaneous )
This is because best_params_ is an argument of GridSearchCV. It is however only created (and accessible) once you run .fit method.