Efficiently counting occurrences of a word list in a column using R

If I have a word list, how do I efficiently count the number of occurrences of these words in a dataset?

An example:

set.seed(123) 
df_data <- data.frame(   
   data_strings = sample(c("tom smith", "smith jim", "sam sam", "ted", "xxx"), 10, replace = TRUE)
)

df_names <- data.frame(
   names = c("tom", "jim", "sam", "ted", "yyy")
)

That is:

> df_data
   data_strings
1       sam sam
2       sam sam
3     smith jim
4     smith jim
5       sam sam
6           xxx
7           ted
8     tom smith
9     smith jim
10      sam sam

and

> df_names
  names
1   tom
2   jim
3   sam
4   ted
5   yyy

I can do this with str_count from the stringr package:

library(stringr)
library(tictoc)
tic()
df_data$counts <- as.vector(sapply(
  paste(df_names[,"names"], collapse='|'), 
  str_count, 
  string=df_data$data_strings
))
toc()

This produces the desired result:

> df_data
   data_strings counts
1       sam sam      2
2       sam sam      2
3     smith jim      1
4     smith jim      1
5       sam sam      2
6           xxx      0
7           ted      1
8     tom smith      1
9     smith jim      1
10      sam sam      2

However, since my real data contains millions of rows, and my word list is also in the millions. This turns out to be a very inefficient way yo get the result. How can I speed it up? I tried to utilize more cores with the parallel package, but it finishes in the same time (it only uses one core although I tell it to use more than one). I'm on windows so I cannot test mclapply(). The parallel seems to be working correctly, since I can get it to use more cores on other examples.

library(stringr)
library(parallel)
library(tictoc)

cl <- makeCluster(4, type = "PSOCK")
tic()
df_data$counts <- as.vector(parSapply(
  cl = cl,
  paste(df_names[,"names"], collapse='|'),
  FUN=str_count, 
  string=df_data$data_strings
))
toc()
stopCluster(cl)

What other approaches could I try? Something with data.tables? Can the paste inside the apply be done differently?