Handling missing values isn't too hard in R. You just need to remember what they represent: unknown data. If you have a collection with an unknown value, you can't know which of the values is smallest. To ignore missing values, you have to explicitly write the code. This lets you, R, and anyone reading your code know what the code assumes.
A lot of functions, including min(), come with an na.rm = TRUE argument.
vals <- c(A = 5, B = 6, C = NA, D = 3)
vals
# A B C D
# 5 6 NA 3
min(vals, na.rm = TRUE)
# [1] 3
Also, I'd recommend using which.min for your code. It returns which element in a vector is the smallest value. It always ignores missing values.
pars <- c("parameterA", "parameterB", "parameterC", "parameterD")
pars[which.min(vals)]
# [1] "parameterD"