Note that, in the interaction net literature, it is pretty common to introduce graphical notation without corresponding textual counterparts. See the works of Lafont, Lamping, Asperti, Guerrini, and many others [1]. (However, the textual calculus can be absolutely crucial for formal proof.)

The absence of proofs and benchmarks undermines the paper for me as well. I find it absolutely critical to demonstrate how the approach works in comparison with already established software, such as BOHM [2].

Parallel beta reduction not being elementarily recursive is not a thing to address, nor a thing one can address. Lamping's abstract algorithm already performs the minimal amount of work to reduce a term -- one cannot improve it further.

From my understanding, the paper aims to optimize the bookkeeping overhead present in earlier implementations of optimal reduction. However, as I said, the absence of formal/empirical evidence of correctness and extensive benchmarks makes the contributions of the paper debatable.

[1] https://github.com/etiamz/interaction-net-resources

[2] https://github.com/asperti/BOHM1.1