Can you afford to miss a count? - the SN74LV8154 has special logic to avoid missing counts - while you read the outputs they hold a snapshot of the true counter which continues to run in the background. For multiple channels each SN74LV8154 can be 2 separate 16 bit counters, you could parallel up all of them and use the shift register chain to read all the bytes in 4 stages (it has 4 output selects).
Read the datasheet about that protocol.
What you don't do is reset the counters - that will lose counts, you take successive readings from each counter and subtract the previous reading to get the new count.