I have been trying to come up with a FA that accepts odd 1's and ends with 101, also leading zeros are allowed. So far I have this.
Problem is as you can see it fails with the string 000010000100010101 which has odd number of 1's and ends with 101. I tried to combine odd 1's and 101 string FA to get this but it doesn't work. So either I am combining them wrong or I shouldn't combine them at all.
In other words, your language consists of strings with an odd number of $1$ followed by $101$ (because $101$ does not change the "oddity" of the number of $1$s). So you do not need to run two automata in parallel, but rather can run them sequentially.
In your start state the number of $1$s is even, add another one for an odd number of $1$s. From the latter one, add a (non-deterministic) transition reading $1$ to the corresponding state in the automaton for $101$, i.e. its state after reading the first $1$.
If this description is too rough, I can provide more detail.